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 🙂

**Answer**

From your definition of the absolute value, establish first $|x| = \max\{x,-x\}$ and $\pm x ≤ |x|$.

Then you can use

\begin{align*}

a + b &≤ |a| + b ≤ |a| + |b|,\quad\text{and}\\

-a – b &≤ |a| -b ≤ |a| + |b|.

\end{align*}

**Attribution***Source : Link , Question Author : ivan , Answer Author : k.stm*