Pointwise convergence of absolute value function

Question

Suppose that $f_1, f_2,..$ be a sequence of function from $\mathbb{R} \to \mathbb{R}$.

Suppose also that the sequence $f_n$ converges pointwise to $f$.

Is it true that the sequence $|f_n|$ also converges pointwise to $|f|$ ? Could you please provide a proof ?

P/s:
I understand that to prove the statement (if it is true), we need to prove that for every $x \in \mathbb{R}$, for every $\epsilon > 0$, there exists a number $N > 0$ such that $||f_n(x)| - |f(x)|| < \epsilon $ for all $n > N$.

However, I don't know how to make appear the absolute value $|f_n(x)|$ as well as $|f(x)|$ from the hypothesis of pointwise convergence of $f_n$.

Answer 1

Hint:

The reverse triangle inequality states that, if $x,y\in \mathbb{R}$ then: $$ \left||x|-|y|\right| \leq |x-y|. $$

Can you apply this to your question?