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I was trying to go through the proof of how to construct the exponential function and got stuck at the last part.

Define exp$(x)=\lim_{n\rightarrow\infty}(1+\frac{x}{n})^n$. Define $e=$ exp$(1)$. I have managed to show that for any rational number $q$, exp$(q)=e^q$. I have also managed to show that exp is a continuous function. I tried to show that for any real number $x$, we have exp$(x)=e^x$ via the following argument:

Let $(q_n)$ be a sequence of rational numbers that converge to $x$. Now,

exp$(x)$ = exp$(\lim_{n\rightarrow\infty}q_n)$ = $\lim_{n\rightarrow\infty} ($exp $q_n)$ = $\lim_{n\rightarrow\infty} e^{q_n}$ = $e^x$

My question is, I wrote the final equality based on instinct instead of relying on a proof. I tried looking for a proof online but did not succeed. The best I got was a paper https://www.iejme.com/download/from-powers-to-exponential-function-5774.pdf that says a proof of the last equality involves the Bernoulli inequality. Would appreciate any help to plug in the gaps for the final equality.

KHOOS
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I would justify it like this: Let $\{ q_n \}$ be a sequence of rational numbers such that $q_n \rightarrow x$. Then we have that $exp(x) = \lim_{n \to \infty} exp(q_n) $ (this is true by continuity of exp). But we also know that $\lim_{n \to \infty} exp(q_n) = \lim_{n \to \infty} e^{q_n} = e^x$ (by the continuity of $e^x$)

Rick Does Math
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