Let $\langle f_n \rangle$ be sequence of equi-continuous real-valued functions on $\mathbb R$ such that $f_n(0)=0$ for every $n$. Does $\langle f_n\rangle$ have a converging subsequence?
I found that even equi-continuity condition and the value at $x=0$ does not guarantee the uniform convergence, as there is a counter-example of $f_n (x) = x/n$, which does not converge uniformly, but do the conditions of the question guarantee pointwise convergence of some subsequence?