\(A=\frac{2}{2\sqrt{1}}+\frac{2}{2\sqrt{2}}+...+\frac{2}{2\sqrt{80}}\)
\(=\frac{2}{\sqrt{1}+\sqrt{1}}+\frac{2}{\sqrt{2}+\sqrt{2}}+...+\frac{2}{\sqrt{80}+\sqrt{80}}\)
\(A>\frac{2}{\sqrt{1}+\sqrt{2}}+\frac{2}{\sqrt{2}+\sqrt{3}}+...+\frac{2}{\sqrt{80}+\sqrt{81}}\)
\(A>\frac{2\left(\sqrt{2}-1\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+\sqrt{1}\right)}+\frac{2\left(\sqrt{3}-\sqrt{2}\right)}{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)}+...+\frac{2\left(\sqrt{81}-\sqrt{80}\right)}{\left(\sqrt{81}-\sqrt{80}\right)\left(\sqrt{81}+\sqrt{80}\right)}\)
\(A>2\left(\sqrt{2}-\sqrt{1}\right)+2\left(\sqrt{3}-\sqrt{2}\right)+...+2\left(\sqrt{81}-\sqrt{80}\right)\)
\(A>2\left(\sqrt{81}-\sqrt{1}\right)=16\) (đpcm)