I am working on the following exercise:
Show that a sequence $$(a_1, 0, 0, \ldots), (0, a_2, 0, \ldots), \ldots, (0, \ldots, 0, a_n, 0, \ldots), \ldots$$ in $\mathbb{R}^\infty$ has a limit if and only if it has only finitely many nonzero terms.
I don't think this is correct. Take $a_i = \frac{1}{i}$. Clearly this sequence converges to 0 but there are not finitely many nonzero terms. Am I missing something here?