sequences and series - Bernoulli's representation of Euler's number, i.e $e=lim limits_{xto infty} left(1+frac{1}{x}right)^x$

Possible Duplicates:
Finding the limit of $n/\sqrt[n]{n!}$
How come such different methods result in the same number, $e$?

I've seen this formula several thousand times:

$$e=\lim_{x\to \infty} \left(1+\frac{1}{x}\right)^x$$

I know that it was discovered by Bernoulli when he was working with compound interest problems, but I haven't seen the proof anywhere. Does anyone know how to rigorously demonstrate this relationship?

EDIT: Sorry for my lack of knowledge in this, I'll try to state the question more clearly. How do we prove the following?

$$\lim_{x\to \infty} \left(1+\frac{1}{x}\right)^x = \sum_{k=0}^{\infty}\frac{1}{k!}$$

Answer

From the binomial theorem

$$\left(1+\frac{1}{n}\right)^n = \sum_{k=0}^n {n \choose k} \frac{1}{n^k} = \sum_{k=0}^n \frac{n}{n}\frac{n-1}{n}\frac{n-2}{n}\cdots\frac{n-k+1}{n}\frac{1}{k!}$$

but as $n \to \infty$, each term in the sum increases towards a limit of $\frac{1}{k!}$, and the number of terms to be summed increases so

$$\left(1+\frac{1}{n}\right)^n \to \sum_{k=0}^\infty \frac{1}{k!}.$$