I have a sequence of functions $f_n:[0,1] \rightarrow \mathbb{R}^n$ such that $f_n$ is uniformly bounded, i.e. $\|f_n\|\leq M$ with $M$ independent of $n$.
Is it true that that the sequence $(f_n)$ converges pointwise to some function $f$, $\lim f_n(x)=f(x)$ for all $x \in [0,1]$ ?
Best Answer
It is not true. For example, let the range just be $\mathbb{R}$, and take $f_n(x)=(-1)^n.$ This is a uniformly bounded sequence, but it does not converge pointwise. You can extend this to $\mathbb{R}^m$ in the obvious way.