Rudin Theorem 4.2 proof question

Question

Refering to the question Rudin 4.2 definition of a limit of a function, I dont have enough reputation to add a comment.

I have the same question. Is'nt the proof in $\impliedby\ $ direction a logical fallacy? If $P \implies\ Q$ then all we know more is $\neg Q \implies\ \neg P$ called modus tollens.

Further $\neg P\implies\ \neg Q$ is called "Denying the antecedent, or Inverse error or Fallacy of the inverse"

Isnt that what Rudin is doing here? So the proof should'nt start by assuming (4) is false, but that (5) is false.

Then from $\neg (5) \land (6) \implies\ \neg (4)$ or $ (5) \land \neg (6) \implies\ \neg (4)$ or (since logical and) $\neg (5) \land \neg (6) \implies\ \neg (4)$ hence a contradiction.

Answer 1

No. The two statements claimed to be equivalent are:

• $P$: $\lim_{x \rightarrow p}f(x) = q$
• $Q$: for every sequence $(p_n)$ in $E$ such that $p_n \neq p$ and $\lim_{n\rightarrow \infty}p_n = p$, $\lim_{n \rightarrow \infty}f(p_n) = q$.

He shows first that $P \implies Q$ and then that $\neg P \implies \neg Q$. The latter is equivalent to $Q \implies P$, which completes the proof that $P \iff Q$.