How could someone conceive of using this inequality for this proof? In James Stewart's Calculus: Early Transcendentals ($8$e), problem $90$ of Section $11.1$ asks us to prove that the sequence
$$a_n=\left(1+\frac{1}{n}\right)^n$$
has a limit. It is divided into five sub-problems, the first of which asks us to show that if $a$ and $b$ are real numbers with $0\leq a<b$, then
$$\frac{b^{n+1}-a^{n+1}}{b-a}<(n+1)b^n$$
After completing all five sub-problems, I can see why proving this inequality was one of them: it's useful for proving that $a_n$ satisfies the hypotheses of the Monotonic Sequence theorem. What I don't see is how one could conceive of using this particular inequality in this proof.
Let's say I wanted to prove that $a_n$ has a limit, but didn't have the guidance of the book. Are there any at-a-glance properties possessed by the sequence that would lead me to consider using the above inequality in my proof? If not, how could I guide my thinking in that direction?
 A: It's speculation on my part, but surely whoever wrote the proof knew that geometric series inequality first, found an argument that used the inequality as an intermediate step, then broke the argument into a sequence of steps for students. The mysterious step, where they "found an argument", is often hard to articulate, and in any case it's frequently not written down. In this particular case, I don't think it's a priori clear that particular inequality "should" be used, so I don't think your literal question has a great answer.
To illustrate this rather common situation more clearly, here's an alternate proof I just came up with, along with discussion afterwards about what's "really going on".
Alternate proof, exercise outline version:

*

*Show that $\int_1^\infty \int_t^\infty \frac{1}{x(1+x)^2}dx\,dt \leq \frac{1}{2}$. (Hint: $\frac{1}{1+x} < \frac{1}{x}$.)


*Let $L(t) = \int_1^t 1/x\,dx$ and let $f(t) = t L(1+1/t)$ for $t>0$. Show that $\lim_{t \to \infty} f'(t) = 0$.


*Show that $f''(t) = -\frac{1}{t(1+t)^2}$. In particular, conclude that $f''(t) < 0$ for $t > 0$.


*Use (2) and (3) to show that $f'(t) = \int_t^\infty \frac{1}{x(1+x)^2}dx > 0$ for all $t > 0$. In particular, conclude that $f(t)$ is increasing for all $t>0$.


*Use (1) and (4) to show that $f(t)$ is bounded as $t \to \infty$. In particular, conclude that $\lim_{t \to \infty} f(t)$ exists.


*Show that $a L(b) = L(b^a)$ for all $a, b > 0$. (Hint: use $u=x^a$.) In particular, conclude that $\lim_{t \to \infty} L((1+1/t)^t)$ exists.


*Show that $L(t)$ is strictly increasing for $t>0$ and conclude that it has a continuous inverse.


*Use (6) and (7) to conclude that $\lim_{t \to \infty} (1+1/t)^t$ exists.
What's really going on?
My first thought was that $(1+1/n)^n$ mixes bases and powers, making things hard to analyze. So, we should fix the base and use exponentials instead: $(1+1/n)^n = \exp(f(n))$ where $f(n) = n\log(1+1/n)$. For this problem, we need to show $f(n)$ monotonically increases and is bounded. Discretizing things by using integer $n$ is needlessly restrictive, so let's replace $n$ with $t$ and analyze $f(t) = t \log(1+1/t)$.
The most obvious thing to do is show that $f'(t) > 0$. By the product rule, $f'(t)$ will still have a log left over (by itself!), and in any case will be "mixed" with a rational function of $t$, and it probably won't be obvious that the result is positive. But, if we differentiate again, the log will die, so without even doing the actual calculations, I knew $f''(t)$ would be a rational function. Such rational functions are frequently "obviously" positive when they need to be, and anyway I know a lot about rational functions, so I figured I'd be able to get it to work. Indeed such is the case here in step (3).
Once we've got $f''(t) < 0$, it's just tying up loose ends with the Fundamental Theorem of Calculus to get in this case that $f'(t) > 0$. To get that $f(t)$ is bounded as $t \to \infty$ requires glancing at the polynomial growth rates of the integrals involved for computing "$f(\infty)$" in terms of the rational function $f''(t)$; the necessary bound is step (1).
As I was typing it out, it occurred to me that assuming the existence of logarithms is probably not fair for a problem that is basically defining $e$, so I added the stuff about $L(t)$ like steps (6) and (7) to derive its basic properties without making the whole argument circular.

A lot of learning more advanced mathematics boils down to being able to understand what the author didn't write as much as what they did write, or in any case being able to come up with your own ways to fill gaps. Gauss was famous for "covering his tracks". Modern standards on the amount of "derivation" (as opposed to "proof") vary depending on the discipline and author.
If you're really interested in this stuff, I'd suggest a book more focused on rigor than Stewart--Rudin's Principles of Mathematical Analysis comes to mind.
