Exercise on sequences using epsilon
You have$$\sqrt2^{\,n}=\bigl(1+\left(\sqrt2-1\right)\bigr)^n\geqslant n\left(\sqrt2-1\right)$$and therefore$$\left(\frac1{\sqrt2}\right)^n\leqslant\frac1{n\left(\sqrt2-1\right)}.$$So, take$$m_\varepsilon=\left\lceil\frac1{n\left(\sqrt2-1\right)\varepsilon}\right\rceil.$$