I've looked up several related questions, but they do not answer what I am looking for. Please give link if this is a duplicate.

What I eventually want to know is why

$$\lim_{n\to\infty}\left(1+\frac1n\right)^n$$

converges to a finite value.

**I mean, we have to know that the limit converges before we can call it a constant, right?**

I know that $f(n)=\left(1+\dfrac1n\right)^n$ is strictly increasing, but how do we know it is bounded above?

Also, I tried using the binomial theorem to write

$$\lim_{n\to\infty}\left(1+\frac1n\right)^n=\lim_{n\to\infty}\left[\sum_{k=0}^n\frac{n!}{n^k k! (n-k)!}\right],\tag{1}$$

but I am unsure how to prove that the sum converges for $n\to\infty,$ because the general term depends on both $n$ and $k.$

Here is what I **think** will work.

Let $a_{n,k}=\dfrac{n!}{n^k k! (n-k)!}.$ Then $a_{n+1,k+1}=\dfrac{(n+1)!}{(n+1)^{k+1} (k+1)! (n-k)!}.$

Using the Ratio Test, we get

$$\lim_{n\to\infty\atop k\to\infty}\left|\frac{a_{n+1,k+1}}{a_{n,k}}\right|=\lim_{n\to\infty\atop k\to\infty}\left[\frac{1}{k+1}\left(\frac{n}{n+1}\right)^k\right].$$

where the limit on the RHS is less than 1.

That may take care of the case where $n\in\mathbb{N},$ so I suppose that (1) can be generalized using "Newton's generalised binomial theorem."