calculus - Calculate the following: $lim limits_{n to infty} sqrt[n]{e^n+(1+frac{1}{n})^{n^2}}$

So here it is: $$\lim_{n \to \infty} \sqrt[n]{e^n+(1+\frac{1}{n})^{n^2}}~~~(n \in \mathbb{N})$$

I tried to use the Squeeze theorem like so:
$e \leq \sqrt[n]{e^n+(1+\frac{1}{n})^{n^2}} \leq e \sqrt[n]{2}$

And if that is true, the limit is simply $e$.

Only that, for it to be true, I first need to prove that $(1+\frac{1}{n})^{n^2} \leq e^n$. So how do I prove it?

Alternatively, how else can I calculate the limit?
I prefer to hear about the way I tried, though.

Thanks