I've been studying calculus and I understand what sufficient and necessary conditions are, but is it true that every sufficient condition is also a necessary one?
For example:
Let $\sum{a_n}$ be a convergent series. If the series is convergent, then $a_n\to 0$. From this we can conclude that necessary condition for series convergence is that $a_n$ must be a zero sequence. A sufficient condition for series convergence is that its partial sum must also converge. My question is, is this condition also necessary? Are there convergent series whose partial sum isn't convergent?
Best Answer
Let $A$ be a proper subset of $B$ and let $B$ be a proper subset of $C$.
Then $x\in A$ is sufficient (but not necessary) for $x\in B$ and $x\in C$ is necessary (but not sufficient) for $x\in B$.
It is handsome to keep this picture/Venn diagram in mind.