418531 ภาคต้น 2552/โจทย์ปัญหาการพิสูจน์ II/เฉลยข้อ 6

ข้อย่อย 1

ก่อนอื่นเราจะนิยามฟังก์ชัน min และ max สำหรับค่าสองค่าก่อน ดังนี้

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 876232399
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:38 GMT


Sensitive client information

IP address: 158.108.32.49

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

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 886738716
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:39 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

และ

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

ข้อย่อย 2

ก่อนเราจะพิสูจน์ข้อความในโจทยฺ์ เราจะพิสูจน์ lemma ต่อไปนี้ก่อน

lemma 1: ให้ $a\,$ และ

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 870302190
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:19:00 GMT


Sensitive client information

IP address: 158.108.32.49

$b\,$ 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 885135062
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:40 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 864048573
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:19:01 GMT


   Sensitive client information

   IP address: 158.108.32.49

   $a>b\,$ 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 852946565
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:40 GMT


   Sensitive client information

   IP address: 158.108.32.49

   $-a<-b\,$ ดังนั้น $\mathrm {max2} (-a,-b)=-b=-\mathrm {min2} (a,b)\,$
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 890310142
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:41 GMT


   Sensitive client information

   IP address: 158.108.32.49

   $a\leq b\,$ เราได้ว่า 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 887166074
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:41 GMT


   Sensitive client information

   IP address: 158.108.32.49

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

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

(Base Case) $n=1\,$ เราจะได้ว่า

(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 894633476
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:43 GMT


Sensitive client information

IP address: 158.108.32.49

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

$\max(-a_{1},a_{2},\ldots ,-a_{n+1})\,$		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 883401513 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:44 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 891718941 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:44 GMT Sensitive client information IP address: 158.108.32.49 $\mathrm {max2} (-\min(a_{1},a_{2},\ldots ,a_{n}),-a_{n+1})\,$
		$-\mathrm {min2} (\min(a_{1},a_{2},\ldots ,a_{n}),a_{n+1})\,$	(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 860378605 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:45 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 891163583
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:46 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{1},a_{2}\,$ และ 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 851768610
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:47 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

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 868567047
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:47 GMT


Sensitive client information

IP address: 158.108.32.49

$A=\mathrm {max2} (a_{1},a_{2})\,$ 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 858938727
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:47 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{1}\leq A,a_{2}\leq A,b_{1}\leq B,$ 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 890933419
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:48 GMT


Sensitive client information

IP address: 158.108.32.49

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

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 862830849
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:48 GMT


Sensitive client information

IP address: 158.108.32.49

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

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 874364849
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:49 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{1}+b_{1}\,$ 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 871415060
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:49 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{2}+b_{2}\,$ เราจึงได้ว่า

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 865881135
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:50 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

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 858677295 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:51 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 865978816 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:51 GMT Sensitive client information IP address: 158.108.32.49 $\leq \,$		(สมมติฐาน)
	$\leq \,$	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 892863966 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:19:15 GMT Sensitive client information IP address: 158.108.32.49 $\mathrm {max2} (\max(a_{1},a_{2},\ldots ,a_{n}),a_{n+1})+\mathrm {max2} (\max(b_{1},b_{2},\ldots ,b_{n}),b_{n+1})\,$	(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 894240407 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:52 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 $\max(a_{1}+b_{1},a_{2}+b_{2},\ldots ,a_{n}+b_{n})\leq \max(a_{1},a_{2},\ldots ,a_{n})+\max(b_{1},b_{2},\ldots ,b_{n})$ สำหรับจำนวนจริง $a_{1},a_{2},\ldots ,a_{n}$ ใดๆ

ข้อย่อย 4

ก่อนเราจะพิสูจน์ข้อความในโจทยฺ์ เราจะพิสูจน์ lemma ต่อไปนี้ก่อน

lemma 3: ให้ 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 892079078
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:53 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 851866314
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:53 GMT


Sensitive client information

IP address: 158.108.32.49

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

พิสูจน์ (lemma 3): ให้ $\alpha =\mathrm {min2} (a_{1},a_{2})\,$ และให้ เราได้ว่า 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 856448666
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:54 GMT


Sensitive client information

IP address: 158.108.32.49

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

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 894797276
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:54 GMT


Sensitive client information

IP address: 158.108.32.49

$\mathrm {min2} (a_{1}+b_{1},a_{2}+b_{2})\,$ มีค่าเท่ากับ

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 859561545
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:55 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{1}+b_{1}\,$ 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 895909997
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:19:21 GMT


Sensitive client information

IP address: 158.108.32.49

$a_{2}+b_{2}\,$ เราจึงได้ว่า

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 869483811
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:55 GMT


Sensitive client information

IP address: 158.108.32.49

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

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

(Base Case) $n=1\,$ เราจะได้ว่า

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 870629843
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:55 GMT


Sensitive client information

IP address: 158.108.32.49

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

(Induction Case) ให้ n เป็นจำนวนเต็มบวกและสมมติให้

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 863718814 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:56 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 861296415 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:19:24 GMT Sensitive client information IP address: 158.108.32.49 $\mathrm {min2} (\min(a_{1}+b_{1},a_{2}+b_{2},\ldots ,a_{n}+b_{n}),a_{n+1}+b_{n+1})\,$
	$\geq \,$		(สมมติฐาน)
	$\geq \,$		(ใช้ lemma 3)
	$=\,$	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 871809561 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:57 GMT Sensitive client information IP address: 158.108.32.49 $\min(a_{1},a_{2},\ldots ,a_{n+1})+\min(b_{1},b_{2},\ldots ,b_{n+1})\,$

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 869287829
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 22 Jun 2026 07:18:58 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