Let f :  ]0;1[ $\longrightarrow \mathbb{R} $, uniformly continuous on ]0;1[. I want to show that f is bounded. My idea is to show that $lim_{x \rightarrow 0+} f(x)$ and $lim_{x \rightarrow 1-} f(x)$ exists (take a sequence ($x_n) \subset ]0;1[ $ ; $\forall n \in \mathbb{N}  x_n &gt;0$ ;  $lim_{n \rightarrow + \infty} x_n = 0$ then by using that f is uniformly continuous we have that (f($x_n$)) is Cauchy and as a result the limit $lim_{x \rightarrow 0+} f(x)$ exists. Same reasoning for the other limit) . Hence we can extend by continuity f in 0 and 1, so f is uniformly continuous on [0;1] and is bounded. I think this reasoning is to simple and wonder what is the problem in it. Thank you by advance for your help.
I need to prove that if $f: (0,1) \rightarrow \mathbb{R}$ is Uniformly continuous then it is  bounded.   Thank you.
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