# Proof of triangle inequality

I understand intuitively that this is true, but I’m embarrassed to say I’m having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated 🙂

From your definition of the absolute value, establish first $$|x| = \max\{x,-x\}$$ and $$\pm x ≤ |x|$$.
\begin{align*} a + b &≤ |a| + b ≤ |a| + |b|,\quad\text{and}\\ -a – b &≤ |a| -b ≤ |a| + |b|. \end{align*}