Continuity of limit of continuous functions implies uniform convergence? 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 ?

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 $x\geq 1/n$. 
