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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.

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    Perhaps the $\sup$ in (1) should be attained? Otherwise, take $f_n = 0$, this satisfies all the conditions, but obviously there is no point for which any $|f_n|$ equals $a>0$.2012-06-08
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    @copper.hat: Thank you for your comment, but I'm not given this information. In fact, I'm given that $|f_{n}(x)|\leq M$ for all $x\in \mathbb R$, and all $n\geq 1$, and I think this means that $\sup_{\mathbb R}|f_{n}|\leq M$, I'm right!?2012-06-08
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    You are correct.2012-06-08
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    I excluded the possibility of having zero functions (because this will not effect my problem). Does it change anything now?2012-06-08

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