请用反证法证明收敛数列的极限是唯一的 用反证法证明极限的唯一性时，为什么取ε=(b-a)/2
\u7528\u53cd\u8bc1\u6cd5\u8bc1\u660e\u6570\u5217Xn\u6536\u655b\uff0c\u90a3\u4e48\u4ed6\u7684\u6781\u9650\u552f\u4e00\u3002\u5176\u4e2d\u4e3a\u4f55\u53d6€=b-a/2\uff1f\u56e0\u4e3a\u8fd9\u662fa\u548cb\u7684\u8ddd\u79bb\u7684\u4e00\u534a\uff0c\u5f53\u7136\u4f60\u4e5f\u53ef\u4ee5\u53d6\u5b83\u76841/3,1/4\u2026\u2026\uff0c\u53ea\u8981\u662f|b-a|\u7684\u6b63\u7684\u5e38\u6570\u500d(\u8fd9\u4e2a\u5e38\u6570\u5c0f\u4e8e1)\u5373\u53ef
\u5177\u4f53\u539f\u56e0\u5982\u4e0b\uff1a
\u8bc1\u660e\u5982\u4e0b\uff1a
\u5047\u8bbe\u5b58\u5728a\uff0cb\u4e24\u4e2a\u6570\u90fd\u662f\u51fd\u6570f(x)\u5f53x\u2192x\u3002\u7684\u6781\u9650\uff0c\u4e14a<b\uff0c\u6839\u636e\u6781\u9650\u7684\u67ef\u897f\u5b9a\u4e49\uff0c\u6709\u5982\u4e0b\u7ed3\u8bba\uff1a
\u4efb\u610f\u7ed9\u5b9a\u03b5>0\uff08\u8981\u6ce8\u610f\uff0c\u8fd9\u4e2a\u03b5\u662f\u5bf9a\uff0cb\u90fd\u6210\u7acb\uff09\u3002
\u603b\u5b58\u5728\u4e00\u4e2a\u03b41>0\uff0c\u5f530<\u4e28x-x\u3002\u4e28<\u03b41\u65f6\uff0c\u4f7f\u5f97\u4e28f\uff08x\uff09-a\u4e28<\u03b5\u6210\u7acb\u3002
\u603b\u5b58\u5728\u4e00\u4e2a\u03b42>0\uff0c\u5f530<\u4e28x-x\u3002\u4e28<\u03b42\u65f6\uff0c\u4f7f\u5f97\u4e28f\uff08x\uff09-b\u4e28<\u03b5\u6210\u7acb\u3002
\u4e0a\u9762\u7684\u4e0d\u7b49\u5f0f\u53ef\u4ee5\u7b49\u4ef7\u53d8\u6362\u4e3aa-\u03b5<f\uff08x\uff09<a+\u03b5\u2460\u548cb-\u03b5<f\uff08x\uff09<b+\u03b5\u2461\u3002
\u4ee4\u03b4=min{\u03b41\uff0c\u03b42}\uff0c\u5f530<\u4e28x-x\u3002\u4e28<\u03b4\u65f6\u3002\u2460\uff0c\u2461\u4e24\u4e2a\u4e0d\u7b49\u5f0f\u540c\u65f6\u6210\u7acb\u3002
\u56e0\u4e3a\u2460\uff0c\u2461\u4e24\u4e2a\u4e0d\u7b49\u5f0f\u540c\u65f6\u6210\u7acb\uff0c\u6240\u4ee5\u2460\u5f0f\u53f3\u7aef\u5fc5\u5b9a\u5927\u4e8e\u6216\u7b49\u4e8e\u2461\u5f0f\u5de6\u7aef\u3002
\u5373\uff1ab-\u03b5\u2264a+\u03b5\uff0c\u79fb\u9879\u5f97\uff1a(b-a)/2\u2264\u03b5,\u56e0\u4e3a(b-a)/2\u662f\u4e00\u4e2a\u786e\u5b9a\u5927\u5c0f\u7684\u6b63\u6570\uff0c\u6240\u4ee5\u8fd9\u4e2a\u7ed3\u8bba\u4e0e\u6781\u9650\u7684\u5b9a\u4e49\uff1a\u03b5\u53ef\u4ee5\u4efb\u610f\u5c0f\u77db\u76fe\uff0c\u6240\u4ee5\u5047\u8bbe\u4e0d\u6210\u7acb\uff0c\u56e0\u6b64\u4e0d\u5b58\u5728a\uff0cb\u4e24\u4e2a\u6570\u90fd\u662ff\uff08x\uff09\u7684\u6781\u9650\uff0c\u9664\u975ea=b\u77db\u76fe\u624d\u4e0d\u4f1a\u51fa\u73b0\u3002
\u5018\u82e5\u662fx\u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\u7684\u552f\u4e00\u6027\u8bc1\u660e\u53ef\u4ee5\u53c2\u770b\u9ad8\u6570\u4e66\u6570\u5217\u6781\u9650\u552f\u4e00\u6027\u8bc1\u660e\uff0c\u8bc1\u6cd5\u5b8c\u5168\u4e00\u6837\u3002
\u8bc1\u6bd5\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u53cd\u8bc1\u6cd5\u7684\u903b\u8f91\u539f\u7406\u662f\u9006\u5426\u547d\u9898\u548c\u539f\u547d\u9898\u7684\u771f\u5047\u6027\u76f8\u540c\u3002
\u5b9e\u9645\u7684\u64cd\u4f5c\u8fc7\u7a0b\u8fd8\u7528\u5230\u4e86\u53e6\u4e00\u4e2a\u539f\u7406\uff0c\u5373\uff1a
\u539f\u547d\u9898\u548c\u539f\u547d\u9898\u7684\u5426\u5b9a\u662f\u5bf9\u7acb\u7684\u5b58\u5728\uff1a\u539f\u547d\u9898\u4e3a\u771f\uff0c\u5219\u539f\u547d\u9898\u7684\u5426\u5b9a\u4e3a\u5047\uff1b\u539f\u547d\u9898\u4e3a\u5047\uff0c\u5219\u539f\u547d\u9898\u7684\u5426\u5b9a\u4e3a\u771f\u3002
\u82e5\u539f\u547d\u9898\uff1a
\u4e3a\u771f
\u5148\u5bf9\u539f\u547d\u9898\u7684\u7ed3\u8bba\u8fdb\u884c\u5426\u5b9a\uff0c\u5373\u5199\u51fa\u539f\u547d\u9898\u7684\u5426\u5b9a\uff1ap\u4e14¬q\u3002
\u4ece\u7ed3\u8bba\u7684\u53cd\u9762\u51fa\u53d1\uff0c\u63a8\u51fa\u77db\u76fe\uff0c\u5373\u547d\u9898\uff1ap\u4e14¬q \u4e3a\u5047\uff08\u5373\u5b58\u5728\u77db\u76fe\uff09\u3002
\u4ece\u800c\u8be5\u547d\u9898\u7684\u5426\u5b9a\u4e3a\u771f\u3002
\u518d\u5229\u7528\u539f\u547d\u9898\u548c\u9006\u5426\u547d\u9898\u7684\u771f\u5047\u6027\u4e00\u81f4\uff0c\u5373\u539f\u547d\u9898\uff1ap⇒q\u4e3a\u771f\u3002
\u8bef\u533a\uff1a
\u5426\u547d\u9898\u4e0e\u547d\u9898\u7684\u5426\u5b9a\u662f\u4e24\u4e2a\u4e0d\u540c\u7684\u6982\u5ff5\u3002
\u547d\u9898\u7684\u5426\u5b9a\u53ea\u9488\u5bf9\u539f\u547d\u9898\u7684\u7ed3\u8bba\u8fdb\u884c\u5426\u5b9a\u3002\u800c\u5426\u547d\u9898\u540c\u65f6\u5426\u5b9a\u6761\u4ef6\u548c\u7ed3\u8bba\uff1a
\u539f\u547d\u9898\uff1ap⇒q\uff1b
\u5426\u547d\u9898\uff1a¬p⇒¬q\uff1b
\u9006\u5426\u547d\u9898\uff1a¬q⇒¬p\uff1b
\u547d\u9898\u7684\u5426\u5b9a\uff1ap\u4e14¬q\u3002
\u539f\u547d\u9898\u4e0e\u5426\u547d\u9898\u7684\u771f\u5047\u6027\u6ca1\u6709\u5fc5\u7136\u8054\u7cfb\uff0c\u4f46\u539f\u547d\u9898\u548c\u539f\u547d\u9898\u7684\u5426\u5b9a\u5374\u662f\u5bf9\u7acb\u7684\u5b58\u5728\uff0c\u4e00\u4e2a\u4e3a\u771f\u53e6\u4e00\u4e2a\u5fc5\u7136\u4e3a\u5047\u3002
\u5df2\u77e5\u67d0\u547d\u9898\uff1a\u82e5A\uff0c\u5219B\uff0c\u5219\u6b64\u547d\u9898\u67094\u79cd\u60c5\u51b5\uff1a
1.\u5f53A\u4e3a\u771f\uff0cB\u4e3a\u771f\uff0c\u5219A⇒B\u4e3a\u771f\uff0c\u5f97¬B⇒¬A\u4e3a\u771f\uff1b
2.\u5f53A\u4e3a\u771f\uff0cB\u4e3a\u5047\uff0c\u5219A⇒B\u4e3a\u5047\uff0c\u5f97¬B⇒¬A\u4e3a\u5047\uff1b
3.\u5f53A\u4e3a\u5047\uff0cB\u4e3a\u771f\uff0c\u5219A⇒B\u4e3a\u771f\uff0c\u5f97¬B⇒¬A\u4e3a\u771f\uff1b
4.\u5f53A\u4e3a\u5047\uff0cB\u4e3a\u5047\uff0c\u5219A⇒B\u4e3a\u771f\uff0c\u5f97¬B⇒¬A\u4e3a\u771f\uff1b
\u2234\u4e00\u4e2a\u547d\u9898\u4e0e\u5176\u9006\u5426\u547d\u9898\u540c\u771f\u5047\u3002
\u5373\u53cd\u8bc1\u6cd5\u662f\u6b63\u786e\u7684\u3002
\u5047\u8bbe¬B\uff0c\u63a8\u51fa¬A\uff0c\u5c31\u8bf4\u660e\u9006\u5426\u547d\u9898\u662f\u771f\u7684,\u90a3\u4e48\u539f\u547d\u9898\u4e5f\u662f\u771f\u7684\u3002
\u4f46\u5b9e\u9645\u63a8\u8bc1\u7684\u8fc7\u7a0b\u4e2d\uff0c\u63a8\u51fa¬A\u662f\u76f8\u5f53\u56f0\u96be\u7684\uff0c\u6240\u4ee5\u5c31\u8f6c\u5316\u4e3a\u4e86\u63a8\u51fa\u4e0e¬A\u76f8\u540c\u6548\u679c\u7684\u5185\u5bb9\u5373\u53ef\u3002\u8fd9\u4e2a\u76f8\u540c\u6548\u679c\u5c31\u662f\u4e0eA\uff08\u5df2\u77e5\u6761\u4ef6\uff09\u77db\u76fe\uff0c\u6216\u662f\u4e0e\u5df2\u77e5\u5b9a\u4e49\u3001\u5b9a\u7406\u3001\u5927\u5bb6\u90fd\u77e5\u9053\u7684\u4e8b\u5b9e\u7b49\u77db\u76fe\u3002
limxn=b
a<b
任意ε>0,存在N1>0,当n>N1时
|xn-a|<ε
任意ε>0,存在N2>0,当n>N2时
|xn-b|<ε
不妨令ε=(b-a)/2
当N=max{N1,N2}时
有|xn-a|<ε,有
xn<(b+a)/2
|xn-b|<ε，有
(b+a)/2<xn
矛盾。
所以
唯一
设：limxn=a，limxn=b，a!=b，
那么：0!=a-b=limxn-limxn=lim(xn-xn)=0，也就是说：0!=0，矛盾了。
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绛旓細a1>0,鈭碼<n+1>=(1/2)(an+1/an)>=1,浜庢槸a<n+1>-an=(1/2)(1-an)(1+an)/an<=0,瀵筺>=2鎴愮珛锛屸埓{an}鏄掑噺鏈変笅鐣岀殑鏁板垪锛鏈夋瀬闄x,浜庢槸 x=(1/2)(x+1/x),x^2=1,x>=1,鈭磝=1,涓烘墍姹傘
