Let $\{f_k(x)\}_{k=1}^\infty$ be a sequence of differentiable functions defined on $[0,1]$. Suppose $f_k(0)=0$ and $|f_k'(x)|\le M$ for all $x\in[0,1]$ and for all $k\ge1$. Prove that after passing to a subsequence, $f_k$ converges uniformly on $[0,1]$ as $k\to\infty$.
In the above question, what is meant by "after passing to a subsequence"? My interpretation is that I am to show that there exists a subsequence which converges uniformly on $[0,1]$ as $k\to\infty$. Is this correct?
Note that I am not asking $why$ we pass to a subsequence. I am asking $what$ passing to a subsequence means.
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$\begingroup$The term "passing to a subsequence" means considering a subsequence instead of the original sequence. However, to avoid making the notation more cumbersome, the subsequence keeps the same name as the original sequence.
For example, consider a sequence $\{a_n\}$, and the subsequence $a_{n_1}, a_{n_2},\ldots$. We could express this as $\{a_{n_i}\}$, but this would be awkward. Since the specific indices of the subsequence don't matter (in some cases), we simply rename the subsequence to have the name of the original sequence.
For the specific problem asked, you are asked to prove not that $\{f_k\}$ converges uniformly, but that there exists some $k_1, k_2,\ldots \subseteq \mathbb{N}$ such that $\{f_{k_i}\}$ converges uniformly.
$\endgroup$ $\begingroup$Since my reputation is at this moment below 50, I cannot add a comment to reply to @zsbrads2, so I am answering in here. Mostly, I would like to point you all towards the comments in Why is "passing to a subsequence" necessary?, where the question is almost answered.
By passing to a subsequence we get rid of the elements that are least convenient. The pertinent subsequence will depend on the specific problem, as well as the techniques involved.
By the way, the statement you want to prove, as it is formulated, is false. Consider the sequence where $f_{2n} = \frac{M}{2}x^2$ and $f_{2n+1} = {M}x$. All of them have their first derivatives bounded by $M$ and are $0$ at $0$, so they satisfy the hypotheses. However, there is no uniform convergence, since the sequence keeps oscillating between these two functions ad infinitum.
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