Suppose converges. Then converges <===> converges.
Should I prove this by contradiction using partial sums of the three series?
What’s confusing is that there is only one case to show here (even though it looks as if not). Without loss of generality, assume one of them converges. Just look at the partial sums to get the final one.
Let , . Suppose and . A fundamental limit property tells us that . Thus converges.