I'm not comfortable with a certain proof that the sequence converges to the number e. The proof uses the fact that 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

which we know converges to e. Also, by comparing terms we see that . So, . What remains is to show which is the part I don't get. The book says to use the sum of the first terms in where .

So,

Now they it says to hold m fixed and let n increase, which gives and therefore .

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.