I have this problem which I think the Mean Value Theorem for continuous functions may apply.
Let $\{f_{n}\}_{n\geq 1}$ be a sequence of non-zero continuous real functions on $\mathbb R$, with the following properties:
(1) $\sup_{x\in \mathbb R}|f_{n}(x)|\leq M$ for some $M>0$, i.e., the sequence is uniformly bounded on $\mathbb R$.
(2) There exists a countable set $W \subset\mathbb R$ such that $\sup_{w\in W}|f_{n}(w)|\to 0$ as $n\to \infty$
Question: Is there an $a\in (0,M)$ on the $y$-axis, such that -for every $n$ - we can find a point, say $x_{n}$ on the $x$-axis with $|f_{n}(x_{n})|=a$?
Note: The set $W$ has no accumulation (limit) point.
Edit: I aded that the functions are nonzero.