Bounded uniformly convergent sequence of functions

Question

${f_n}$ is a sequence of (real-valued) functions uniformly converging to $f$.

Suppose further that $f_n(x)< M$ (The functions $f_n$ are uniformly bounded above by $M$).

Is it true that $f(x) < M$ everywhere?

Answer 1

Let $f(x)=1$, $f_n(x)=1−1/n$ and $M=1$. Answer: no.

But it is true if $<$ is replaced by $\le$, because if $f(x)>M$ then $f_n(x)>M$ for $n$ sufficiently large, contradiction.