Give an example of an equicontinuous sequence of differentiable functions $f_n:[0,1]\to \mathbb{R}$ that is not uniformly bounded, but satisfies $|f'_n(x)|\leq 1$ for all $n\in \mathbb{N}$ and $x\in [0,1]$. Justify whether $\{f_n\}_{n=1}^{\infty}$ has a uniform convergent subsequence on $[0,1]$.
What about $f_n(x)=\frac{\sin(nx)}{n}$? This is equicontinuous , and $|f'_n(x)|=|\cos(nx)|\leq 1$. I guess it has a uniformly convergent subsequence which converges to $0$.
Does anyone have better function?
Best Answer
Let $f_n(x)=n$. Then $f_n'(x)=0$ for all $x$ and the family is clearly equicontinuous. The family is also not bounded, so in particular it is not uniformly bounded.