Could you please suggest a sketch of proof for this problem:
$f: A \to \mathbb{R}$ and $f(x) \gt a$ for every $x$.
Given that $f_k(x)$ converges pointwise to $f(x)$.
Prove that there exists a natural number $N$ such that $f_k(x) \gt a$ for every $k \gt N$. ($i.e. x$ is fixed)
It seems very intuitive but I just can't find a rigorous way to go from the definition of poitwise limit to the conclusion.
Thank you for your help!
This can be proved using the definition of pointwise convergence. For a fixed $x$, $\exists~\epsilon>0$ such that $f(x)-\epsilon>a$. For that $\epsilon>0, ~~~\exists~N \in \mathbb{N}$ such that for all $k>N$, $|f_k(x)-f(x)|<\epsilon$ holds. That implies $f_k(x)>f(x)-\epsilon>a$ for all $k>N$.