I have been given a sequence of real valued continuous functions $(f_1,f_2,...,f_n)$, and a real valued sequences $(x_1,x_2,...,x_n)$, where $x_n$ converges to $x$. Also $f$ is a continuous real valued function.
Then if $f_n$ converges to $f$ uniformly on $\mathbb R$, I think I have shown that this implies that $f_n(x_n)$ converges to f(x).
But does this hold when $f_n$ converges to $f$ pointwise on $\mathbb R$? I am trying to come up with a counterexample, but I haven't had any luck so far.. Does anyone know of such an example, or does pointwise convergence imply the same as uniform convergence?