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 $n \longrightarrow \infty$)?  

Suppose otherwise. Then there is an an interval $(x-a,x+a)$ such that $x_n\notin(x-a,x-a)$ for infinitely many $n$'s. Define$$f(y)=\left\{\begin{array}{ll}x+a&\text{ if }y\geqslant x+a\\y&\text{ if }y\in(x-a,x+a)\\x-a&\text{ if }y\leqslant x-a.\end{array}\right.$$Then $\bigl|f(x_n)-x\bigr|\geqslant a$ for infinitely many $n$'s and therefore $\lim_nx_n\not\to x$.
