I am struggling to prove the following theorem:
Prove that:
$$\lim_{x \to \infty} f(x) = L$$ iff $$\lim_{x \to 0^+} \frac{1}{x} = L$$
on $A: (0, \infty)$.
My idea was that that if we let $x, \frac{1}{x}$ be a subset of $A$, then we can construct a sequence of $(f(x)) < (\frac{1}{x})$. Then, since $(f(\frac{1}{x}))$ is bounded and decreasing, it is monotone. By the monotone convergence theorem, that sequence converges.
If a sequence converges, then its subsequence converges as well. Therefore $f(x)$ converges to the same thing as $\frac{1}{x}$.
I know that the proof I have given is wrong, but I am struggling to understand how to properly construct it.
