Prove the following statement or find a counter example:Every convergent sequence that is uniformly bounded on a compact set of real numbers contains a uniformly convergent subsequence.
proof on a uniformly convergent subsequence
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sequences-and-series
convergence-divergence
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0@Danikar That makes sense. I mostly wanted the OP to clarify the question. I don't think there's a reason to be vague :) – 2012-02-18
2 Answers
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Define $f_n: [0, 1] \to \mathbb{R}$ by $f_n(x) = \frac{x^2}{x^2 + (1 - nx)^2}.$
$|f_n(x)| \le 1$, so uniformly bounded. And $f_n \to f = 0$. But for any subsequence $\{f_{n_k}\}$ we have $f_{n_k}(1/n_k) = 1$, so no subsequence converges uniformly.
See Rudin, Principles of Mathematical Analysis, Chapter 7 page 156.
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Consider the sequence of functions $f_n(x)=x^n$ on the interval $[0,1]$. For $x\in[0,1)$, we have $f_n(x)\rightarrow0$ and for $x=1$, we have $f_n(x)\rightarrow 1$. The limiting function of this sequence is not even continuous!