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 🙂

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

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