Pointwise convergence in increasing functions

Question

If $f_{n}$ are a sequence of real valued strictly increasing functions that converges pointwise to $f$, is it necessary that $f$ be strictly increasing? Looks fine, but I can't see why. Thanks in advance!

Answer 1

No. Take $$f_n(x)=\frac x{2^n}$$ For each $n$, $f_n$ is strictly increasing, but $\lim_{n\to\infty}f_n(x)=0$ for all $x$. All we can say is that the limit is monotone increasing.

Answer 2

This question is part of the following "problem": if $(u_n) \to l \in \Bbb R$ and $ (\forall n, u_n > 0) $ does this implies that $l >0$? In general, the answer is no (take $u_n = \frac{1}{n} \to 0$). Here, you can take $f_n(x) = \frac{x}{n}$ for every $x \in \Bbb R^+$. We see that each $f_n$ is strictly increasing, $f_n \to 0$ but the $0$ function isn't strictly increasing.