Example of a sequence that follows certain properties

Question

I'm trying to find a sequence, $(f_n)_{n \in \mathbb{N}}\subset E$, with $E = C([0,1])$, that respects the following proprieties:

$1.) \Vert f_n \Vert_\infty \leq 1 \ \forall_{n \in \mathbb{N}}$

$2.) (f_n)$ has not a convergent subsequence.

After some work I came up with this one:

$$ f_n(x) = \frac{x}{1 + \vert \sin(nx) \vert} $$

but I am not sure

Thank you for your help.

Answer 1

I don't know whether that sequence works or not, but one that works is$$f_n(x)=\begin{cases}2nx&\text{ if }x\in\left[0,\frac1{2n}\right]\\2-2nx&\text{ if }x\in\left(\frac1{2n},\frac1n\right)\\0&\text{ otherwise.}\end{cases}$$The sequence $(f_n)_{n\in\Bbb N}$ converges pointwise to the null function. Therefore, if some subsequence was convergent in $\left(C\bigl([0,1]\bigr),d_\infty\right)$, it would converge to the null function. But, for each $n\in\Bbb N$, the range of $f_n$ is $[0,1]$, and therefore $\|f_n-0\|_\infty=1$.