\(\displaystyle e = \lim_{n \to \infty} (1 + \frac{1}{n})^n\)

I'll lay it out.

**1:**\(\displaystyle e^1 = \lim_{n \to \infty} (1 + \frac{1}{n})^n = \lim_{n \to \infty} ( \frac{n + 1}{n})^n\)

**2:**\(\displaystyle e^-1 = \frac{1}{e^1} = \lim_{n \to \infty} ( \frac{n }{n + 1})^n\)

**3:**Set

**x = n + 1**so that

**n = x - 1**

**4:**\(\displaystyle e^-1 =\lim_{x \to \infty} (\frac{x - 1}{x})^x-1 =\lim_{x \to \infty} (\frac{x - 1}{x})^x(\frac{x - 1}{x})^1\) (the latex messes up but that -1 is in the exponent in the first part here, as is obvious from the second part!).

**5:**\(\displaystyle = \lim_{x \to \infty} (1 - \frac{1}{x})^x(1 - \frac{ 1}{x})^1 = \lim_{x \to \infty} (1 - \frac{1}{x})^x \cdot 1\)

Now the proof tells me to return back to 'n' instead of x.

**6:**\(\displaystyle \lim_{x \to \infty} (1 - \frac{ 1}{x})^x \Rightarrow \ \lim_{n \to \infty} (1 - \frac{ 1}{n + 1})^n + 1 \) (where the +1 at the end here is in the exponent!)

**7:**\(\displaystyle \lim_{n \to \infty} (1 - \frac{ 1}{n + 1})^n + 1 = \lim_{n \to \infty} (\frac{n + 1 - 1}{n + 1})^n(\frac{n + 1 - 1}{n + 1})\)

**8:**\(\displaystyle \lim_{n \to \infty} (\frac{n }{n + 1})^n(\frac{n }{n + 1})\)

But the proof I've seen has substituted in the 'n' on it's own, not the 'n + 1' as I have & I don't understand, is there something I'm missing? Obviously the thing I've ended up with is strange, it looks similar to the original e^-1 but with that extra fraction.