Continuity of limit of continuous functions implies uniform convergence?

Question

We know that uniform convergence of continuous functions implies the continuity of the limit. I'm wondering if the inverse is true: if a sequence of continuous functions $\{f_n\}$ converges to continuous function $f$, then is it true that $\{f_n\}$ converges uniformly to $f$.

My attempt is that for any $x$ in the domain and $\epsilon$, there exists a $a$ such that $|f(x)-f_n(x)|\le|f(x)-f(a)|+|f(a)-f_n(a)|+|f_n(a)-f_n(x)|$. But I can't estimate the distance between $|f(a)-f_n(a)|$. Any tips ?

Answer 1

Hint: Construct for $n\geq 1$ a continuous function $f_n$ for which $f_n(0)=0$, $f_n(1/2n)=1$ and $f_n(x)\equiv 0$ for $x\geq 1/n$.

Answer 2

$f_n(x) = n^2x^n(1-x) \to 0$ pointwise in $[0,1].$ But note $f_n(1-1/n) = n^2(1-1/n)^n(1/n) \to \infty.$ Thus $\sup_{[0,1]}|f_n - 0|$ does not go to $0$ (far from it), so the convergence is not uniform.