Can someone verify this claim:
If a sequence $(f_n)$ do not converge uniformly, then no subsequence
of $(f_n)$ converge uniformly
I saw this in a proof, where $f_n(x) = x^n$ on $x \in [0,1]$, the claim it is well known that the above sequence is not uniformly convergent, therefore no subsequence of $(f_n)$ is uniformly convergent
Best Answer
The claim isn't true in general. Suppose that $f_n\to 0$ pointwise but not uniformly, and define $g_{2n}=f_n$, $g_{2n+1}=0$. Then $g_n$ does not converge to zero uniformly, but a subsequence does.