Problem 2

Show that the inequality

\sum_{i=1}^n\sum_{j=1}^n\sqrt{|x_i-x_j|}\le\sum_{i=1}^n\sum_{j=1}^n\sqrt{|x_i+x_j|}

holds for all real numbers

x_1,x_2,\ldots,x_n

Step 1 of 5: Verify the base cases

n=1

and

n=2

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In plain words

Small cases anchor the induction and are checked by direct computation.

n=1:\ 0\le 2\sqrt{|x_1|}

Detailed analysis

For $n=1$ both sides equal $0$ trivially (or reduce to $0\le 2\sqrt{|x_1|}$ once diagonal terms are separated), and for $n=2$ a short case check on the sign of $x_1x_2$ confirms the inequality; these anchor an induction on $n$ .