convergence divergence - difference of two series converges, but when does each converge?

My question below was stupid. I was meaning to ask the question in a more specific context. (Without mentioning the context, it's a stupid question!) Please ignore this question.

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Consider two sequences $a_n$ and $b_n$ in a compact set. Assume $|a_n-b_n| \leq \frac{C}{n}$. Then $a_n - b_n$ converges to zero; however, $a_n$ and $b_n$ may not converge.

My question is: again, consider two sequences $a_n$ and $b_n$ in a compact set. Assume $|a_n-b_n| \leq \frac{C}{n^2}$. Do $a_n$ and $b_n$ converge? (My intuition says yes since $\sum \frac{1}{n}$ diverges, but $\sum \frac{1}{n^2}$ converges) How do I show it? Or is it NOT true?