<s>[INST] Prove that $\sqrt[n]{n} < (1 + \frac{1}{\sqrt{n}})^2$ for all n in the naturals. I need to prove that $\sqrt[n]{n} < (1 + \frac{1}{\sqrt{n}})^2$ for all n in the naturals.
I started by using Bernoulli's inequality: 
$(1+\frac{2}{\sqrt{n}}) < (1 + \frac{1}{\sqrt{n}})^2$
I can say that:
$(1+\frac{2}{\sqrt{n}}) = (1+\frac{2\sqrt{n}}{n})$
I can also subtract the one and divide by 2 on the left side without changing the inequality (because it makes it even smaller): $(\frac{\sqrt{n}}{n}) < (1 + \frac{1}{\sqrt{n}})^2$
But now I am stuck...
 [/INST] $$\begin{align}
\left(1+{1\over\sqrt n}\right)^{2n}
&=\left(1+{2\over\sqrt n}+{1\over n}\right)^n\\
&\gt\left(1+{2\over\sqrt n} \right)^n\\
&\ge1+{n\choose1}{2\over\sqrt n}+{n\choose2}\left(2\over\sqrt n\right)^2\\
&\gt1+0+2(n-1)\\
&=2n-1\\
&\ge n
\end{align}$$
</s>