Let $f: \mathbb{N} \to \mathbb{N} $ be a bijective function, the series $ \displaystyle \sum_ {n = 0}^{\infty} a_ {n} $ converges if and only if the series $ \displaystyle \sum_ {n = 0}^{\infty} a_ {f (n) }$ is convergent.
my attempt:
Since $f$ is bijective, the only thing it does is to permute the set $\mathbb{N}$, so if any of the series converges when the elements of $\mathbb{N}$ are perrmuted, their partial sums would continue to converge, therefore the proposition is true.
However in my class they said that the statement is false so I tried to find a counterexample, but I could not find it. I am really confused. Any help that makes me see that the proposition is true or false would be greatly appreciated.
