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If $n = xm$ and $n \rightarrow \infty$, then $m \rightarrow \infty$?

Did Stewart prove the titular result, also underlined in green in the image below, either by himself or as an exercise, in *Calculus Early Transcendentals* or in the normal version *Calculus*?
I attempted the proof, but I got nonplussed. $\infty = \lim\limits_{n \rightarrow \infty} n = \lim\limits_{n \rightarrow \infty} xm $. Now what? The limit is for $n$, but my variables are $x$ and $m$.
>56. Show that $\lim\limits_{n \rightarrow \infty} (1 + \dfrac xn)^n = e^x$ for any $x > 0$.
>[![James Stewart, *Calculus Early Transcendentals* 7 ed 2011. p. 223.][1]][1]
>
> *Transcription of image follows below:*
> **56.** Let $m = n/x$. Then $n = xm$, and as $n \to \infty$, $m \to \infty$.
>
> Therefore, $\displaystyle \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n = \lim_{m \to \infty} \left( 1 + \frac{1}{m} \right)^{mx} = \left[ \lim_{m \to \infty} \left( 1 + \frac{1}{m} \right)^m \right]^x = e^x$ by Equation 6.
>
> <sup>James Stewart, *Calculus Early Transcendentals*, 7 ed. (2011) p. 223.</sup>
By the way, if I was writing the solution, I would've started with the game plan: rewriting $(1 + x/n)$ in the form $(1 + 1/\mathrm{something})^{\mathrm{something}}$ in order to apply $e = \lim\limits_{something \rightarrow \infty} (1 + 1/\mathrm{something})^{\mathrm{something}}$. It's superfluous to change variable to $m$. Just rewrite $(1 + x/n)$ as $\left(1 + \dfrac{1}{\frac nx}\right).$
But if you pined to change variable, I would've commenced with defining $\dfrac xn$ as $\dfrac 1 m$ which is more intuitive than "let $m = n/x$", because only $\dfrac xn$ shows up explicitly in the question. The question doesn't manifest $n/x$.
[1]: https://i.stack.imgur.com/WYuLX.jpg

Did Stewart prove the titular result, also underlined in green in the image below, either by himself or as an exercise, in *Calculus Early Transcendentals* or in the normal version *Calculus*?
I attempted the proof, but I got nonplussed. $\infty = \lim\limits_{n \rightarrow \infty} n = \lim\limits_{n \rightarrow \infty} xm $. Now what? The limit is for $n$, but my variables are $x$ and $m$.
>56. Show that $\lim\limits_{n \rightarrow \infty} (1 + \dfrac xn)^n = e^x$ for any $x > 0$.
><details>
><summary>Image of below context</summary>
> <img src="https://i.stack.imgur.com/WYuLX.jpg">
>
></details>
>
> *Transcription of image follows below:*
> **56.** Let $m = n/x$. Then $n = xm$, and as $n \to \infty$, $m \to \infty$.
>
> Therefore, $\displaystyle \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n = \lim_{m \to \infty} \left( 1 + \frac{1}{m} \right)^{mx} = \left[ \lim_{m \to \infty} \left( 1 + \frac{1}{m} \right)^m \right]^x = e^x$ by Equation 6.
>
> <sup>James Stewart, *Calculus Early Transcendentals*, 7 ed. (2011) p. 223.</sup>
By the way, if I was writing the solution, I would've started with the game plan: rewriting $(1 + x/n)$ in the form $(1 + 1/\mathrm{something})^{\mathrm{something}}$ in order to apply $e = \lim\limits_{something \rightarrow \infty} (1 + 1/\mathrm{something})^{\mathrm{something}}$. It's superfluous to change variable to $m$. Just rewrite $(1 + x/n)$ as $\left(1 + \dfrac{1}{\frac nx}\right).$
But if you pined to change variable, I would've commenced with defining $\dfrac xn$ as $\dfrac 1 m$ which is more intuitive than "let $m = n/x$", because only $\dfrac xn$ shows up explicitly in the question. The question doesn't manifest $n/x$.
[1]: https://i.stack.imgur.com/WYuLX.jpg

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