TITLE: How does this proof of Theorem 1 in Spivak's Calculus work?
QUESTION [10 upvotes]: Hello I just started Spivak's Calculus and I've come across something I don't really understand. What is trying to be proved is
$$ 
|a+b|\leq|a|+|b|\,.
$$
The proof is based on the observation that $|a|=\sqrt{a^2}$. So:
$$
\begin{align*}
(|a+b|)^2 = (a+b)^2 &= a^2+2ab+b^2 \\
&\leq a^2+2|a|\times|b|+b^2 \\
&= |a^2|+2|a|\times|b|+|b^2| \\
&= (|a|+|b|)^2 \\
\end{align*}
$$
Now what I don't understand is the change in relationship from line 1 to 2 ($= to \leq$) and from line 2 to 3 ($\leq to =$).
I attach the page because it's clearer there (I highlighted the proof in yellow). Actual proof from the book

REPLY [10 votes]: The lines are read left to right and top to bottom. So the symbol ≤ on line 2 relates the last expression of line 1 to the next expression on line 2. Similarly, the symbol = on line 3 relates the last expression of line 2 to the next expression on line 3. The same thing could be written in one line as follows:
\begin{equation}
\begin{split}
(|a+b|^2) = (a+b)^2 &= a^2+2ab+b^2
\leq a^2+2|a|\times|b|+b^2
= |a^2|+2|a|\times|b|+|b^2|
= (|a|+|b|)^2,
\end{split}
\end{equation}
but it is easier on the eyes to write each new expression on a new line.