[分析] 一致連續題二

[playmypig (玩我豬)]

題目:Suppose that f(x) is uniformly continuous in an open interva; (a,b). Prove that f(x) is bounded in (a,b).

我的解答:
Suppose f is not bounded on (a,b), there exists a sequence {x_n} in (a,b) such that |f(x_n)|>n for n=1, 2, 3, …Since {x_n} in (a,b), it has a convergent subsequence {x_(n_k ) }  in (a,b).
Since {x_(n_k ) } is convergent, then {x_(n_k ) }  is A Cauchy Sequence. f({x_(n_k ) }) is also a Cauchy Sequence in R.
However, since |f(x_n)|>n for all n=1,2,3,…f({x_(n_k ) })cannot be a Cauchy Sequence. This is a contradiction. So, f is bounded on (a,b).

真的有勞各位了, 我想問問我的答題有沒有錯誤呢?謝謝.


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推 jacky7987   :可以簡單的用你上面的第二題的結果
→ jacky7987   :f可以extend到[a,b],[a,b] is cpt, 所以有極值定理
→ playmypig   :謝謝回答,我做好了