Let $(f_n)$ be a sequence of monotonically increasing functions which converges pointwise to $f$ . Show that f must also be monotonically increasing

Question

Let $(f_n)$ be a sequence of monotonically increasing functions which converges pointwise to $f$ . Show that f must also be monotonically increasing

My attempt: Now assume $f_n$ converges to $f$ uniformly and monotonically increasing

suppose $x<y$

$|f(x)-f(y)| = |f(x)-f_n(x)|+|f_n(x)-f_n(y)|+|f_n(y)-f(x)|\\ \leq |f_n(x)-f(x)|+|f_n(y)-f_n(x)|+|f_n(y)-f(y)|$

in which $n$ is an arbitrary natural number. Now fix $x$ and $y$, so we have two sequence of numbers on both sides of the inequality.

So take $n \to ∞ $ we will see that $f(x) \leq f(y)$.

but is this true for even $f_n$ is pointwise?

Answer 1

Take $x<y$. Then for any $n\in \mathbb{N}$, we have $$ f_{n}(x)<f_n(y) $$ if the sequence is strictly monotonically increasing.

Since the inequality holds for any $n$, we have $$ f(x)=\lim_{n\to \infty}f_n(x)\leq\lim_{n\to \infty}f_n(y)=f(y) $$ note that the strict inequality may have been lost.