Suppose that {${f_n}$} is a sequence of functions on $[a,b]$ that has the property that, for every convergent sequence {$x_n$} of numbers in $[a,b]$,$$\lim\limits_{n \to \infty} f_n(x_n)=0$$ Show that {$f_n$} converges uniformly to the zero function on $[a,b]$.
My attempt:
Assume toward a contradiction that {$f_n$} does not converge uniformly to the zero function on $[a,b]$. Then, there exists some $\epsilon$ $>0$ and some $N$ such that if $n>N$, then $|f_n(x)-0|\geq\epsilon$. Since$$\lim\limits_{n \to \infty} f_n(x_n)=0$$ we know that $|f_n(x)-0|<\epsilon$. This is a contradiction, and thus {$f_n$} must converge uniformly to the zero function on $[a,b]$.