1. a and b are integers, therefore a-b is an integer. An integers square is always greater then or equal to 0.

\(\displaystyle{0}\le{\left({a}-{b}\right)}^{{2}}\)

\(\displaystyle{\left({a}-{b}\right)}^{{2}}={a}^{{2}}-{2}{a}{b}+{b}^{{2}}\)

\(\displaystyle{0}\le{a}^{{2}}-{2}{a}{b}+{b}^{{2}}\)

\(\displaystyle{2}{a}{b}\le{a}^{{2}}-{2}{a}{b}+{b}^{{2}}+{2}{a}{b}\)

\(\displaystyle{2}{a}{b}\le{a}^{{2}}+{b}^{{2}}\)

2. Since a, b and c are integers, any difference between them is also an integer. An integers square is always greater then or equal to 0.

\(\displaystyle{0}\le{\left({a}-{b}\right)}^{{2}}+{\left({b}-{c}\right)}^{{2}}+{\left({c}-{a}\right)}^{{2}}\)

\(\displaystyle{0}\le{a}^{{2}}-{2}{a}{b}+{b}^{{2}}+{b}^{{2}}-{2}{b}{c}+{c}^{{2}}+{c}^{{2}}-{2}{a}{c}+{a}^{{2}}\)

\(\displaystyle{2}{a}{b}+{2}{b}{c}+{2}{a}{c}\le{2}{a}^{{2}}+{2}{b}^{{2}}+{2}{c}^{{2}}\)

\(\displaystyle{a}{b}+{b}{c}+{a}{c}\le{a}^{{2}}+{b}^{{2}}+{c}^{{2}}\)

\(\displaystyle{0}\le{\left({a}-{b}\right)}^{{2}}\)

\(\displaystyle{\left({a}-{b}\right)}^{{2}}={a}^{{2}}-{2}{a}{b}+{b}^{{2}}\)

\(\displaystyle{0}\le{a}^{{2}}-{2}{a}{b}+{b}^{{2}}\)

\(\displaystyle{2}{a}{b}\le{a}^{{2}}-{2}{a}{b}+{b}^{{2}}+{2}{a}{b}\)

\(\displaystyle{2}{a}{b}\le{a}^{{2}}+{b}^{{2}}\)

2. Since a, b and c are integers, any difference between them is also an integer. An integers square is always greater then or equal to 0.

\(\displaystyle{0}\le{\left({a}-{b}\right)}^{{2}}+{\left({b}-{c}\right)}^{{2}}+{\left({c}-{a}\right)}^{{2}}\)

\(\displaystyle{0}\le{a}^{{2}}-{2}{a}{b}+{b}^{{2}}+{b}^{{2}}-{2}{b}{c}+{c}^{{2}}+{c}^{{2}}-{2}{a}{c}+{a}^{{2}}\)

\(\displaystyle{2}{a}{b}+{2}{b}{c}+{2}{a}{c}\le{2}{a}^{{2}}+{2}{b}^{{2}}+{2}{c}^{{2}}\)

\(\displaystyle{a}{b}+{b}{c}+{a}{c}\le{a}^{{2}}+{b}^{{2}}+{c}^{{2}}\)