I want to prove that for $m→∞ ⇒((1+x/m)^m→exp(x))$

My idea is to prove that there is an $m≥n$ so that $$(1+x/m)^m≥\sum\_{j=0}^{n}x^j/j!≥(1+x/n)^n$$

now I would use the binomial theorem to rewrite $(1+x/m)^m, (1+x/n)^n$ as a sum, but I don't know how to prove both inequalities. With the left inequality I simply don't know which $m$ to choose, and in the right inequality I just get horribly stuck with the summations. Any help please?