Let $(f_n)$ be a sequence of real valued continuous functions defined on a complete metric space $X$ such that $f(x) :=$ lim $f_n(x)$ exists for all $x \in X$.

these are problems.

(a) is there a nonempty open subset $O$ and a constant $M < \infty$ such that $\mid f_n(x) \mid < M$ for all $x \in O$ and for all $n \ge 1$?

(b) Given $\epsilon > 0$, is there a nonempty open subset $O$ and an integer $N$ such that $\mid f(x) - f_n(x) \mid < \epsilon$ for all $x \in O$ and for all $n \ge N$ ?

I can prove that $C(X)$ is complete metric space. but I don't know how to choose open set $O$. where do I have to start to solve these problems?

  • 2
    $\begingroup$ Sounds like a Baire Category problem. $\endgroup$
    – PhoemueX
    Jun 11 '16 at 5:20
  • $\begingroup$ The $O$ would come from a certain set having non-empty interior. $\endgroup$ Jun 11 '16 at 5:40

For a fixed $x$, the sequence $(f_n(x))_n$ is convergent, so bounded. So we have $N_x \in \mathbb{N}$ such that $|f_n(x)| \le N_x$ for all $n \ge 1$.

Define $C_N = \{x \in X: \forall n\ge 1: |f_n(x)| \le N \}$, for $N \in \mathbb{N}$. The first paragraph shows $X = \bigcup_{N \in \mathbb{N}} C_N$, as $x \in C_{N_x}$. Can you show that all $C_N$ are closed in $X$? If so, they cannot be all nowhere dense, so one of them has non-empty interior, so contains some non-empty open set $O$...

The second problem is similar. Fix $\varepsilon > 0$.

For every $x \in X$, $(f_n(x))_n$ is convergent, hence Cauchy. So there is $N_x$ again such that for all $n,m \ge N_x$, $|f_n(x) - f_m(x)| \le \frac{\varepsilon}{2}$.

Now define analogous sets $C_N$ and show them to be closed (the $\le$ are important there, as in the previous case).


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