Question about (simple) logic to prove a proposition.

Question

Suppose I have to prove that

1. $A \implies B$ (if a monotone sequence is bounded, then it converges);
2. $\neg A \implies\neg B$ (if a monotone sequence is unbounded, then it diverges).

Assume that I have in hand the result

3. $A \iff B$ (a monotone sequence is bounded iff it converges).

Well, item $1$ directly comes from $3$. But is there any way to use $3$ to deduce $2$. It seens naively obvious to me, but I cannot formally show it, using logic.

Can you help me to work on some argument?

Thanks in advance.

Answer 1

Well, $A\Longleftrightarrow B$ is equivalent to $A\Longrightarrow B\wedge B\Longrightarrow A$.

Moreover, $B\Longrightarrow A$ is equivalent to the contraposition $\neg A\Longrightarrow \neg B$.

Both assertions together should answer your question.

Answer 2

$$(A\iff B)$$

$$\equiv (A\implies B ) \land (B\implies A) $$

$$\equiv (\lnot B \implies \lnot A) \land (\lnot A \implies \lnot B)$$