Existence or non existence of uniformly convergent subsequence

Question

Consider the sequence of functions defined from $[0,1]$ to $\mathbb{R}$, given by $$f_n(x) = \frac{2x^2}{x^2+(1-2nx)^2}~,\quad \text{where} ~~n=1,2,\dots$$ I could conclude that this sequence of functions converges pointwise to zero. But supremum of $|f_n - f|$, (here $f=0$) does not converge to zero, hence convergence is not uniform.

But I'm not able to conclude if there exist a subsequence that converges uniformly or not. Any help would be appreciated.

Thanks in advance.

Answer 1

I think not, because for every $n$ other than $0$, function $f_n(x)$ has a maximum at some point $x_{max}$ such that $$f_n(x_{max})=2$$

Answer 2

The pointwise limit is $f(0)=2, f(x)=0$ for $x \neq 0$. Every subsequence converges to the same limit. Since this limit is not continuous it follows that the convergence is not uniform.