Spivak: Calculus on Manifolds Norms (Problem 1.1) I found it difficult to prove this problem due to the following fact. I want to derive it myself but am having trouble laying out the proof. First, I want to show the following holds:
Show that $$(\sum_{i=1}^n |x_i|)^2= \sum_{i=1}^n x_{i}^2 + 2\sum_{i\neq j}|x_i||x_j|$$
I would also like a little more emphasis on the notation being used for the second sum. Thanks!

An helpful way would be to arrange the terms $|x_1|,...,|x_n|$ in a square array in an increasing order. Regards this square array as a matrix of size $n\times n$ whose $ij$ entry is $|x_i||x_j|$ for $1\leqslant i,j\leqslant n$. The diagonal entries are $|x_i||x_i|=|x_i|^2$ and the off-diagonal entries are given by $|x_i||x_j|$ for $i\neq j$. The total sum of all entries would be 
$$\sum_{i,j}|x_i||x_j|=\sum_{i=1}^n|x_i|^2+\sum_{i>j}|x_i||x_j|+\sum_{i<j}|x_i||x_j|=\sum_{i=1}^n|x_i|^2+2\sum_{i\neq j}|x_i||x_j|$$
where we used $|x_i||x_j|=|x_j||x_i|$ in the very last step (i.e. the matrix is symmetric). This total is also equal to 
$$\Big(\sum_{i=1}^n|x_i|\Big)^2$$
