In a book I am reading the author essentially says:
The condition $\sum\limits_{k\geq 0} a_k^2< \infty $, where $a_k\in R$, implies that $a_k^2 \rightarrow 0$.
This feels wrong to me because intuitively, if I take any series where it is true, and then reorder it to start the indexing from infinity, then the new series would be a counterexample since the initial terms would start near zero and keep growing with the index.
However, that is just my intuition since you can't actually "start the indexing from infinity".
So is the original statement true? Or is there an easy counterexample to prove that it is false?
One misconception in your question is that it's not possible to reorder the sequence so that it "keeps growing with the index".
For example, there's only finitely many terms with value $\ge .00001$, and no matter how you reorder, those finitely many will eventually show up and all the rest of the terms will be $<.00001$.
And then there's only finitely many terms with value $\ge .0000000000000001$, and no matter how you reorder, those finitely many will eventually show up and all the rest will be $<.0000000000000001$.
And so on...
Another way to look at this is that if $\sum_{k \ge 0} a_k^2 = S$, then any finite subset of the terms has sum $\le S$. So no matter how you re-order, and no matter how many of the terms of the re-ordered series you sum up (finitely many terms, that is), that sum is $\le S$.