Let $\{f_n\}_n$ be a sequence of functions which are continuous over $[0, 1]$ and continuously differentiable in $[0, 1]$. Assume that $|f_n(x)| \leqslant 1$ and that $|f_n' (x)| \leqslant 1$ for all $x \in[0, 1]$ and for each positive integer $n$. Pick out the true statement:
- $f_n$ is uniformly continuous for each $n$.
- $f_n$ is a convergent sequence in $C[0, 1]$.
- $f_n$ contains a subsequence which converges in $C[0, 1]$.
From my point of view all options are correct because function derivative is bounded and continuous, so function is uniformly continuous in $[0,1]$. But I'm not sure about my answer because my teacher give me $0$ mark on this question. If anybody help me I would be very thankful to him.
Hint. Take $f_n(x)=(-1)^n$ then $\{f_n\}_n$ is a sequence of of continuous (constant) functions which satisfies the given conditions, that is $|f_n(x)| \leqslant 1$ and $|f_n' (x)| \leqslant 1$ for all $x \in[0, 1]$ and for each positive integer $n$. What may we conclude about option 2.?
Note that $|f_n(x)| \leqslant 1$ and $|f_n' (x)| \leqslant 1$ for all $x \in[0, 1]$ and for all $n\geq 1$ imply that the sequence is equibounded and equicontinuous. For option 3. use Ascoli-Arzelà Theorem.