418531 ภาคต้น 2552/โจทย์ปัญหาการพิสูจน์ II/เฉลยข้อ 6

ข้อย่อย 1

ก่อนอื่นเราจะนิยามฟังก์ชัน min และ max สำหรับค่าสองค่าก่อน ดังนี้

$\mathrm {min2} (a,b)={\begin{cases}a,&a\leq b\\b,&a>b\end{cases}}$

และ

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 270698459
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:19 GMT


Sensitive client information

IP address: 158.108.32.49

$\mathrm {max2} (a,b)={\begin{cases}a,&a>b\\b,&a\leq b\end{cases}}$

จากนั้นเราจึงนิยามฟังก์ชัน min และ max สำหรับค่า n ค่าใดๆ ดังต่อไปนี้

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 265390070
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:20 GMT


Sensitive client information

IP address: 158.108.32.49

$\min(a_{1},a_{2},\ldots ,a_{n})={\begin{cases}a_{1},&n=1\\\mathrm {min2} (\min(a_{1},a_{2},\ldots ,a_{n-1}),a_{n}),&n>1\end{cases}}$

และ

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 260377219
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:20 GMT


Sensitive client information

IP address: 158.108.32.49

$\max(a_{1},a_{2},\ldots ,a_{n})={\begin{cases}a_{1},&n=1\\\mathrm {max2} (\max(a_{1},a_{2},\ldots ,a_{n-1}),a_{n}),&n>1\end{cases}}$

ข้อย่อย 2

ก่อนเราจะพิสูจน์ข้อความในโจทยฺ์ เราจะพิสูจน์ lemma ต่อไปนี้ก่อน

lemma 1: ให้ $a\,$ และ $b\,$ เป็นจำนวนจริงใดๆ แล้ว

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 273384231
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:21 GMT


Sensitive client information

IP address: 158.108.32.49

$\mathrm {max2} (-a,-b)=-\mathrm {min2} (a,b)\,$

พิสูจน์ (lemma 1): การพิสูจน์แบ่งออกเป็นสองกรณี

1. 

   Error

   Too many requests (f061ab2)

   If you report this error to the Wikimedia System Administrators, please include the details below.

   Request served via cp5021 cp5021, Varnish XID 266208944
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:21 GMT


   Sensitive client information

   IP address: 158.108.32.49

   $a>b\,$ เราได้ว่า $-a<-b\,$ ดังนั้น
   

   Error

   Too many requests (f061ab2)

   If you report this error to the Wikimedia System Administrators, please include the details below.

   Request served via cp5021 cp5021, Varnish XID 182871275
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:22 GMT


   Sensitive client information

   IP address: 158.108.32.49

   $\mathrm {max2} (-a,-b)=-b=-\mathrm {min2} (a,b)\,$
2. 

   Error

   Too many requests (f061ab2)

   If you report this error to the Wikimedia System Administrators, please include the details below.

   Request served via cp5021 cp5021, Varnish XID 213173607
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:22 GMT


   Sensitive client information

   IP address: 158.108.32.49

   $a\leq b\,$ เราได้ว่า
   

   Error

   Too many requests (f061ab2)

   If you report this error to the Wikimedia System Administrators, please include the details below.

   Request served via cp5021 cp5021, Varnish XID 265193721
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:22 GMT


   Sensitive client information

   IP address: 158.108.32.49

   $-a\geq -b\,$ ดังนั้น $\mathrm {max2} (-a,-b)=-a=-\mathrm {min2} (a,b)\,$

พิสูจน์ (โจทย์): เราจะทำการพิสูจน์โดยใช้ induction บนตัวแปร $n\,$

(Base Case) $n=1\,$ เราจะได้ว่า

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 223066946
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:23 GMT


Sensitive client information

IP address: 158.108.32.49

$\max(-a_{1})=-a_{1}=-\min(a_{1})\,$

(Induction Case) ให้ n เป็นจำนวนเต็มบวกและสมมติให้ $\max(-a_{1},-a_{2},\ldots ,-a_{n})=-\min(a_{1},a_{2},\ldots ,a_{n})$

ให้

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 271058353
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:24 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{1},a_{2},\ldots ,a_{n+1}$ เป็นจำนวนจริงใดๆ เราได้ว่า

Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 269584441 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:24 GMT Sensitive client information IP address: 158.108.32.49 $\max(-a_{1},a_{2},\ldots ,-a_{n+1})\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 205999838 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:25 GMT Sensitive client information IP address: 158.108.32.49 $=\,$	$\mathrm {max2} (\max(-a_{1},-a_{2},\ldots ,-a_{n}),-a_{n+1})\,$
	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 127233668 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:26 GMT Sensitive client information IP address: 158.108.32.49 $=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 271385380 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:26 GMT Sensitive client information IP address: 158.108.32.49 $\mathrm {max2} (-\min(a_{1},a_{2},\ldots ,a_{n}),-a_{n+1})\,$
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 258804497 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:27 GMT Sensitive client information IP address: 158.108.32.49 $-\mathrm {min2} (\min(a_{1},a_{2},\ldots ,a_{n}),a_{n+1})\,$	(ใช้ lemma 1)
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 264045038 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:27 GMT Sensitive client information IP address: 158.108.32.49 $-\min(a_{1},a_{2},\ldots ,a_{n+1})\,$

ดังนั้นเราสามารถสรุปได้ว่า $\max(-a_{1},-a_{2},\ldots ,-a_{n})=-\min(a_{1},a_{2},\ldots ,a_{n})$ สำหรับจำนวนจริง $a_{1},a_{2},\ldots ,a_{n}$ ใดๆ

ข้อย่อย 3

ก่อนเราจะพิสูจน์ข้อความในโจทยฺ์ เราจะพิสูจน์ lemma ต่อไปนี้ก่อน

lemma 2: ให้

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 252613662
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:28 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{1},a_{2}\,$ และ

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 272894649
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:28 GMT


Sensitive client information

IP address: 158.108.32.49

$b_{1},b_{2}\,$ เป็นจำนวนจริงใดๆ แล้ว

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 261524560
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:29 GMT


Sensitive client information

IP address: 158.108.32.49

$\mathrm {max2} (a_{1}+b_{1},a_{2}+b_{2})\leq \mathrm {max2} (a_{1},a_{2})+\mathrm {max2} (b_{1},b_{2})\,$

พิสูจน์ (lemma 2): ให้ $A=\mathrm {max2} (a_{1},a_{2})\,$ และให้ เราได้ว่า

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 236661715
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:30 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{1}\leq A,a_{2}\leq A,b_{1}\leq B,$ และ $b_{2}\leq B\,$ ฉะนั้น

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 266667499
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:30 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{1}+b_{1}\leq A+B\,$ และ

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 264308911
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:31 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{2}+b_{2}\leq A+B\,$

เนื่องจาก

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 253008368
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:31 GMT


Sensitive client information

IP address: 158.108.32.49

$\mathrm {max2} (a_{1}+b_{1},a_{2}+b_{2})\,$ มีค่าเท่ากับ $a_{1}+b_{1}\,$ หรือไม่ก็

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 263128971
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:32 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{2}+b_{2}\,$ เราจึงได้ว่า $\mathrm {max2} (a_{1}+b_{1},a_{2}+b_{2})\leq A+B=\mathrm {max2} (a_{1},a_{2})+\mathrm {max2} (b_{1},b_{2})\,$

พิสูจน์ (โจทย์): เราจะทำการพิสูจน์โดยใช้ induction บนตัวแปร $n\,$

(Base Case) $n=1\,$ เราจะได้ว่า

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 270501368
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:33 GMT


Sensitive client information

IP address: 158.108.32.49

$\max(a_{1}+b_{1})\leq a_{1}+b_{1}=\max(a_{1})+\max(b_{1})\,$

(Induction Case) ให้ n เป็นจำนวนเต็มบวกและสมมติให้

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 242232148
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:34 GMT


Sensitive client information

IP address: 158.108.32.49

$\max(a_{1}+b_{1},a_{2}+b_{2},\ldots ,a_{n}+b_{n})\leq \max(a_{1},a_{2},\ldots ,a_{n})+\max(b_{1},b_{2},\ldots ,b_{n})\,$

ให้ $a_{1},a_{2},\ldots ,a_{n+1},b_{1},b_{2},\ldots ,b_{n+1}$ เป็นจำนวนจริงใดๆ เราได้ว่า

Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 264242416 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:34 GMT Sensitive client information IP address: 158.108.32.49 $\max(a_{1}+b_{1},a_{2}+b_{2},\ldots ,a_{n+1}+b_{n+1})\,$	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 262800804 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:35 GMT Sensitive client information IP address: 158.108.32.49 $\mathrm {max2} (\max(a_{1}+b_{1},a_{2}+b_{2},\ldots ,a_{n}+b_{n}),a_{n+1}+b_{n+1})\,$
	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 272467592 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:35 GMT Sensitive client information IP address: 158.108.32.49 $\leq \,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 215729657 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:35 GMT Sensitive client information IP address: 158.108.32.49 $\mathrm {max2} (\max(a_{1},a_{2},\ldots ,a_{n})+\max(b_{1},b_{2},\ldots ,b_{n}),a_{n+1}+b_{n+1})\,$	(สมมติฐาน)
	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 254974899 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:36 GMT Sensitive client information IP address: 158.108.32.49 $\leq \,$	$\mathrm {max2} (\max(a_{1},a_{2},\ldots ,a_{n}),a_{n+1})+\mathrm {max2} (\max(b_{1},b_{2},\ldots ,b_{n}),b_{n+1})\,$	(ใช้ lemma 2)
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 258411617 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:37 GMT Sensitive client information IP address: 158.108.32.49 $\max(a_{1},a_{2},\ldots ,a_{n+1})+\max(b_{1},b_{2},\ldots ,b_{n+1})\,$

ดังนั้นเราสามารถสรุปได้ว่า

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 253598964
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:37 GMT


Sensitive client information

IP address: 158.108.32.49

$\max(a_{1}+b_{1},a_{2}+b_{2},\ldots ,a_{n}+b_{n})\leq \max(a_{1},a_{2},\ldots ,a_{n})+\max(b_{1},b_{2},\ldots ,b_{n})$ สำหรับจำนวนจริง $a_{1},a_{2},\ldots ,a_{n}$ ใดๆ

ข้อย่อย 4

ก่อนเราจะพิสูจน์ข้อความในโจทยฺ์ เราจะพิสูจน์ lemma ต่อไปนี้ก่อน

lemma 3: ให้

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 272499610
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:37 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{1},a_{2}\,$ และ

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 265390175
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:38 GMT


Sensitive client information

IP address: 158.108.32.49

$b_{1},b_{2}\,$ เป็นจำนวนจริงใดๆ แล้ว $\mathrm {min2} (a_{1}+b_{1},a_{2}+b_{2})\geq \mathrm {min2} (a_{1},a_{2})+\mathrm {min2} (b_{1},b_{2})\,$

พิสูจน์ (lemma 3): ให้

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 263849246
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:38 GMT


Sensitive client information

IP address: 158.108.32.49

$\alpha =\mathrm {min2} (a_{1},a_{2})\,$ และให้ เราได้ว่า

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 258706445
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:39 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{1}\geq \alpha ,a_{2}\geq \alpha ,b_{1}\geq \beta ,$ และ

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 253956272
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:39 GMT


Sensitive client information

IP address: 158.108.32.49

$b_{2}\geq \beta \,$ ฉะนั้น และ $a_{2}+b_{2}\geq \alpha +\beta \,$

เนื่องจาก มีค่าเท่ากับ หรือไม่ก็

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 252744393
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:40 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{2}+b_{2}\,$ เราจึงได้ว่า $\mathrm {min2} (a_{1}+b_{1},a_{2}+b_{2})\geq \alpha +\beta =\mathrm {min2} (a_{1},a_{2})+\mathrm {min2} (b_{1},b_{2})\,$

พิสูจน์ (โจทย์): เราจะทำการพิสูจน์โดยใช้ induction บนตัวแปร $n\,$

(Base Case) $n=1\,$ เราจะได้ว่า $\min(a_{1}+b_{1})\leq a_{1}+b_{1}=\min(a_{1})+\min(b_{1})\,$

(Induction Case) ให้ n เป็นจำนวนเต็มบวกและสมมติให้

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 215729688
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:40 GMT


Sensitive client information

IP address: 158.108.32.49

$\min(a_{1}+b_{1},a_{2}+b_{2},\ldots ,a_{n}+b_{n})\leq \min(a_{1},a_{2},\ldots ,a_{n})+\min(b_{1},b_{2},\ldots ,b_{n})\,$

ให้ $a_{1},a_{2},\ldots ,a_{n+1},b_{1},b_{2},\ldots ,b_{n+1}$ เป็นจำนวนจริงใดๆ เราได้ว่า

Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 175570066 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:48:02 GMT Sensitive client information IP address: 158.108.32.49 $\min(a_{1}+b_{1},a_{2}+b_{2},\ldots ,a_{n+1}+b_{n+1})\,$		Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 268173483 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:41 GMT Sensitive client information IP address: 158.108.32.49 $\mathrm {min2} (\min(a_{1}+b_{1},a_{2}+b_{2},\ldots ,a_{n}+b_{n}),a_{n+1}+b_{n+1})\,$
	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 223625534 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:48:04 GMT Sensitive client information IP address: 158.108.32.49 $\geq \,$	$\mathrm {min2} (\min(a_{1},a_{2},\ldots ,a_{n})+\min(b_{1},b_{2},\ldots ,b_{n}),a_{n+1}+b_{n+1})\,$	(สมมติฐาน)
	$\geq \,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 238234711 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:47:42 GMT Sensitive client information IP address: 158.108.32.49 $\mathrm {min2} (\min(a_{1},a_{2},\ldots ,a_{n}),a_{n+1})+\mathrm {min2} (\min(b_{1},b_{2},\ldots ,b_{n}),b_{n+1})\,$	(ใช้ lemma 3)

ดังนั้นเราสามารถสรุปได้ว่า

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 244819675
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 08 May 2026 22:48:08 GMT


Sensitive client information

IP address: 158.108.32.49

$\min(a_{1}+b_{1},a_{2}+b_{2},\ldots ,a_{n}+b_{n})\leq \min(a_{1},a_{2},\ldots ,a_{n})+\min(b_{1},b_{2},\ldots ,b_{n})$ สำหรับจำนวนจริง ใดๆ