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 $\pm x ≤ |x|$.
Then you can use
\begin{align*}
a + b &amp;≤ |a| + b ≤ |a| + |b|,\quad\text{and}\\
-a - b &amp;≤ |a| -b ≤ |a| + |b|.
\end{align*}
$|x+y|^2=(x+y).(x+y)
       =(x.x)+2(x.Y)+(y.y)                     
       =|x|^2+2(x.Y)+|y|^2$
from Cauchy-Schwarz inequality,$|x.Y|&lt;=|x||y|$
     $|x+y|^2 &lt;=|x|^2+2|x||y|+|y|^2$
    $ |x+y|^2 &lt;=(|x|+|y|)^2$
taking square root on both sides.
     $|x+y|&lt;=(|x|+|y|)$