If $f(x_n)$ converges for all bounded continuous functions $f$, does then $x_n$ converge? Let $x_n\in \mathbb{R}$ be a sequence and $x\in \mathbb{R}$ such that $f(x_n) \longrightarrow f(x)$ (as $n \longrightarrow \infty$) for all bounded continuous functions $f \colon \mathbb{R} \rightarrow \mathbb{R}$. Do we then have $x_n \longrightarrow x$ (as $n \longrightarrow \infty$)?  

Yes, set
$$f_k(t) = t ,\ \ -k<x<k$$
$$f_k(t) = k , \ \ t\geq k$$
$$f_k(t) = -k , \ \ t\leq k$$
for all k. Then for $|k|>x$, we must have some N where all $|x_n|<k$ for all n>N. So for all such n
$$f_k(x_n) = x_n \rightarrow x$$
