# Rudin 4.2 definition of a limit of a function

Baby Rudin theorem 4.2 presents an alternative definition of a limit:

Suppose $$X, Y$$ are metric spaces, $$E \subset X$$, $$f: X \rightarrow Y$$, $$p$$ is a limit point of $$E$$.

Then $$\lim_{x \rightarrow p}f(x) = q$$ $$\textbf{(4)}$$ if and only if $$\lim_{n \rightarrow \infty}f(p_n) = q$$ $$\textbf{(5)}$$ for every sequence $$(p_n)$$ in $$E$$ such that $$p_n \neq p$$, $$\lim_{n\rightarrow \infty}p_n = p$$ $$\textbf{(6)}$$.

For the proof of $$\impliedby$$, Rudin supposes that $$\textbf{(4)}$$ is false, then finds a sequence in $$E$$ that satisfies $$\textbf{(6)}$$ but not $$\textbf{(5)}$$. I'm not certain as to how this proves the implication, nor where the contradiction is that would close the argument.

• It looks like a proof by contrapositive, rather than contradiction ($p\implies q$ is equivalent to $\neg q\implies \neg p$. – AnyAD Nov 12 '18 at 22:24

That proposition has this structure:$$A\iff(B\implies C).$$In order to prove $$\Longleftarrow$$, Rudin proves that $$\neg A\implies\neg(B\implies C)$$. And, in turn, $$\neg(B\implies C)$$ is equivalent to $$B\wedge\neg C$$.