$\lim\limits_{n\to\infty}$ $(1+ {2\over n})^n$ = $e^2$ If I set N=$2\over n$, the equation becomes $(1+ N)^{2\over N}$, which I can take the natural log of and work down until I reach $e^2$. However, my teacher wants me to use subsequences, starting with $x_n$ = $(1+ {1\over n})^n = e$, to prove this and I am stuck.

You could rewrite this as 
$$\lim_{n \to \infty}\left(1+\frac{1}{n/2}\right)^n = \lim_{n \to \infty}\left(\left(1+\frac{1}{n/2}\right)^{n/2}\right)^2 = \lim_{n\to\infty} (x_n)^2$$
where $x_n=\left(1+\frac{1}{n/2}\right)^{n/2}$. Now as $n \to \infty$, $\frac{n}{2} \to \infty$ as well. Hence $\lim_{n\to\infty} x_n = e$. Thus, bringing the limit inside the square (which we can do since squaring is continuous):
$$\lim_{n \to \infty}\left(1+\frac{1}{n/2}\right)^n = \lim_{n \to \infty}(x_n)^2= \left(\lim_{n\to\infty} x_n \right)^2=e^2.$$
This doesn't "use subsequences," but it makes use of the sequence whose limit you already know, if that was your teacher's intention.
