I'm working through the first chapter of Michael Spivak's Calculus 3rd ed.
Towards the end of the chapter he proves $ |a + b| ≤ |a| + |b| $ using the observation that $|a|= \sqrt{ a^2 }$ when $a$ is $ ≥ 0 $ .
$ |a + b| ≤ |a| + |b| $
$ (|a + b|)^2 = (a + b)^2 $ $= a^2 + 2ab + b^2 $ $ ≤ a^2 + 2|a| |b| + b^2 $ $ = |a|^2 + 2|a| |b| + |b|^2 $ $ = (|a| + |b|)^2 $
I am unsure about what's going on with the equality sign. How does it go from $=$ to $≤$ on line 3 when a and b are changed to their absolute value and back to $=$ again on line 4 when $a^2$ and $b^2$ are changed to their absolute values?