Find a sequence of functions $\{g_n\}$ in $C([0,1])$ so that $\overline{\{g_n\}}$ is compact, but $g_n$ does not converge uniformly.

Question

Consider the space $C([0,1])$ equipped with the uniform norm. Find a sequence of functions $\{g_n\}$ in $C([0,1])$ so that $\overline{\{g_n\}}$ is compact, but $g_n$ does not converge uniformly.

I can't seem to find a sequence of function that satisfy the above statement. I started off by finding a non-uniform convergence sequence and work from there but no luck.

Any help would be greatly appreciated.

https://math.stackexchange.com/questions/1814293/does-pointwise-convergence-of-continuous-functions-on-a-compact-set-to-a-continu — , Mar 21 '19 at 22:03
Do you mean that the closure of the sequence ${g_n}$ is compact in the space $C[0,1]$ with its max-norm? — uniquesolution, Mar 21 '19 at 22:30

score 5 · Accepted Answer · answered Mar 21 '19 at 22:08

5

For each $n\in\mathbb N$ and each $x\in[0,1]$, define $g_n(x)=(-1)^n$.

answered Mar 21 '19 at 22:08

José Carlos Santos

427,504

I might be wrong but I don't think $g_n(x)$ is continuous.. – Pika_2018 Mar 21 '19 at 22:40
4

I could swear that every constant function is continuous. – José Carlos Santos Mar 21 '19 at 22:44
My bad. Thank you! – Pika_2018 Mar 22 '19 at 00:05

Find a sequence of functions $\{g_n\}$ in $C([0,1])$ so that $\overline{\{g_n\}}$ is compact, but $g_n$ does not converge uniformly.

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