Circular argument in proof?

Question

See the part (B). In it, the author "proves" the limit $\lim\limits_{|x|\to\infty}\left(1+\frac 1x\right)^x$. The part concerning $x\to -\infty$ is in the next page but in that, he just takes $y=-x$ and $y\to\infty$ and establishes the limit using the first case, so I didn't include that part in the image.

I have two concerns (questions) regarding this :

(a) Isn't $e$ defined that way? How does one prove a definition ? I can understand proof of a definition if we assume some other definition and show that it's equivalent to this one but I don't see the author doing that.

(b) The author states that "See that when $n\to\infty$ ($n$ going through positive integers only), we have $(1+1/n)^n\to e$ and $(1+1/(n+1))^{n+1}\to e$" and then he concludes his first part of the "proof" using squeeze theorem. Isn't this a circular argument ? How do we know beforehand that those expressions tend to $e$ ?

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Bottom line, do you guys think this is a proof ? I don't think so but I'd like to hear opinions of other MSE users.

Also, I'd like if someone posts a different proof of the same using some other standard definition of $e$, I suppose (probably the infinite series definition would work). Thanks.

Answer 1

The proof is about the limit for real $x$. It relies on a definition of $e$ stated for integer $n$. There is no circular argument and the author is not defining $e$ twice. He is showing that the limit by reals is the same as that by integers.

Technically speaking, the squeeze theorem is invoked after showing that $\lfloor x\rfloor<x<\lfloor x\rfloor+1$ is preserved by monotonocity of the function $\left(1+\dfrac1x\right)^x$.

Answer 2

One can define sequence $$a_n = \left(1+\frac 1 n\right)^n$$ (which is just a function $a\colon\mathbb N\to \mathbb R$)

and one can define function of positive real variable $f\colon\mathbb R_{>0}\to\mathbb R$ $$f(x) = \left(1+\frac 1 x\right)^x$$

Since $(a_n)$ is bounded monotone sequence, it is convergent, and we denote it's limit by $e$ (one possible definition of $e$). But, author is not establishing a limit of a sequence, but a limit of function, and uses convergence of sequence $(a_n)$ and Sandwich theorem to prove that $$\lim_{n\to\infty}a_n = \lim_{x\to\infty}f(x)$$