I'm not comfortable with a certain proof that the sequence $\displaystyle {y_n} = 1 + \frac{1}{{1!}} + \frac{1}{{2!}} + \cdots + \frac{1}{{n!}}$ converges to the number e. The proof uses the fact that $\displaystyle {y_n}$ is an increasing sequence that is bounded (which you show by a greater geometric series). Therefore it is convergent. Then you expand the the sequence

$\displaystyle {x_n} = {\left( {1 + \frac{1}{n}} \right)^n} = 1 + 1 + \frac{1}{{2!}}{\left( {1 - \frac{1}{n}} \right)^{}} + \cdots \frac{1}{{n!}}\left( {1 - \frac{1}{n}} \right)\left( {1 - \frac{2}{n}} \right) \cdots \left( {1 - \frac{{n - (n - 1)}}{n}} \right)$

which we know converges to e. Also, by comparing terms we see that $\displaystyle {x_{n \le }}{y_n}$. So, $\displaystyle {y_n} \ge e$. What remains is to show $\displaystyle {y_n} \le e$ which is the part I don't get. The book says to use the sum of the first $\displaystyle m + 1$ terms in $\displaystyle {x_n}$ where $\displaystyle m < n$.

So,

$\displaystyle = 1 + 1 + \frac{1}{{2!}}{\left( {1 - \frac{1}{n}} \right)^{}} + \cdots \frac{1}{{m!}}\left( {1 - \frac{1}{n}} \right)\left( {1 - \frac{2}{n}} \right) \cdots \left( {1 - \frac{{n - (n - 1)}}{n}} \right) < {x_n} < e$

Now they it says to hold m fixed and let n increase, which gives $\displaystyle {y_m} = 1 + \frac{1}{{1!}} + \cdots \frac{1}{{m!}} \le e$ and therefore $\displaystyle {y_n} \le e$.

But how is it valid to use a finite # of terms? We have to deal with infinite terms, and I tell myself that y may exceed e in that case.