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$\big( \sqrt{2+ \sqrt{2+ \sqrt{ 2+\sqrt{2}}}}~\big)^2\ \textless \ 2^2 \\ 2+ \sqrt{2+ \sqrt{ 2+\sqrt{2}}}\ \textless \ 4 \\\big( \sqrt{2+ \sqrt{ 2+\sqrt{2}}}\big~)^2\ \textless \ 2^2 \\ 2+ \sqrt{ 2+\sqrt{2}}\ \textless \ 4\\\big(\sqrt{2+\sqrt{2}}\big~)^2\ \textless \ 2^2 \\2+\sqrt{2}\ \textless \ 4 \\ (\sqrt{2})^2\ \textless \ 2^2 \\ 2\ \textless \ 4$

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