On limit of point wise convergent sequence of continuous functions on real line

Question

Let $\{f_n\}$ be a sequence of continuous functions on real line which is point wise convergent . Then is it true that for every $c\in \mathbb R$ , the set $\{x \in \mathbb R : \lim_{n\to \infty} f_n(x) \ge c\}$ is a $G_{\delta}$ set (countable intersection of open sets) in $\mathbb R$ ?

Answer 1

Yes, these sets are $G_{\delta}$ sets.

For every $k \in \mathbb{N}$, let

$$s_k(x) = \sup \{ f_n(x) : n \geqslant k\}.$$

Since the sequence is pointwise convergent, $s_k$ is finite everywhere, and since the $f_n$ are continuous, the $s_k$ are lower semicontinuous. Hence for every $a\in \mathbb{R}$, the sets $\{ x : s_k(x) > a\}$ are open. Since

$$A_k :=\{ x : s_k(x) \geqslant c\} = \bigcap_{m = 1}^{\infty} \{ x : s_k(x) > c - 1/m\}$$

we see that each $A_k$ is a $G_{\delta}$. Since $(s_k)$ converges to $f = \lim f_n$ monotonically decreasing, we have

$$\{ x : f(x) \geqslant c\} = \bigcap_{k\in \mathbb{N}} A_k,$$

so the set is a countable intersection of $G_{\delta}$ sets, but those are of course $G_{\delta}$ sets too.

Answer 2

$A=\{x \in \mathbb R : \lim_{n\to \infty} f_n(x) \ge c\}= \{x \in \mathbb R : \limsup_{n \rightarrow \infty}f_n(x) > c \} \cup \{x \in \mathbb{R}|\limsup_{n \rightarrow \infty}f_n(x) =c\}$

Denote $$A_1=\{x \in \mathbb R : \limsup_{n \rightarrow \infty}f_n(x) > c \}$$ $$A_2=\{x \in \mathbb{R}|\limsup_{n \rightarrow \infty}f_n(x) =c\}$$

We have that $\limsup_{n \rightarrow \infty}f_n(x) >c \Rightarrow \forall k \in \mathbb{N},\exists m \geqslant k$ such that $f_m(x)>c$, thus $$A_1= \bigcap_{k=1}^{\infty} \bigcup_{m \geqslant k} \{x \in\mathbb{R}|f_m(x)>c\}$$

From the fact that any union of open sets is open and the fact that because of the continuity of $f_m$ the set $\{x \in\mathbb{R}|f_m(x)>c\}$ is open,we see that $A_1$ is a $G_{\delta}$ set.

Now $\limsup_{n \rightarrow \infty}f_n(x) =c \Rightarrow \forall k \in \mathbb{N}$ ,exist infinitely many numbers $n \in \mathbb{N}$ such that $f_n(x)>c- 1/k$ ,thus $$A_2= \bigcap_{k=1}^{\infty} \bigcup_{ n=1}^ {\infty} \bigcup_{m_n \in \mathbb{N}}f_{m_n}^{-1}((c-1/k, + \infty))$$

In the same way as i mentioned above tou can see that $A_2$ is a $G_{ \delta}$ set.

Therefore $A$ is a $G_{ \delta}$ set a union of two $G_{ \delta}$ sets.