Analysis - Show that if b ∈ R and lim x→a f(x) = b then lim x→a |f(x)| = |b|.

Question

Let $A\subseteq R$ , $f:A\rightarrow R$ be a function and a be an accumulation point of $A$. Show that if $b\in R$ and $\lim_{x\rightarrow a}f\left(x\right)=b$ then $\lim_{x\rightarrow a}\left|f\left(x\right)\right|=\left|b\right|$. Feel like Heine definition is needed but unsure.

Answer 1

Hint: $\left|\left|f\left(x\right)\right|-\left|b\right|\right|\leq\left|f\left(x\right)-b\right|$ by the reverse triangle inequality.