Proving a theorem, what is meant by sufficiency and necessity? I am looking at the proof of a theorem and the proof begins by saying > ...is the proof of the sufficiency part of this theorem so we just need to establish the necessity of the condition. What is the sufficiency part and the necessity part of the theorem?

It is essentially a biconditional, also known as an if and only if.

An "if and only if" statement goes both ways. That is, $p\iff q$ means "if $p$ is true then $q$ is true" **and** "if $q$ is true then $p$ is true."

The statement "$p$ is sufficient for $q$" means "if $p$ is true, then $q$ is true."

The statement "$p$ is necessary for $q$" means that if we don't have $p$, then we don't have $q$. Therefore, if we have $q$, we certainly have $p$. In other words, "$q$ implies $p$."

When we put the two together, a necessary and sufficient condition is the same as an if and only if.