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So we're given a sequence $x_n$ and a sequence $y_n$, both of them being of real numbers. We know that $x_n \rightarrow 0$ and that $y_n$ is bounded. We need to prove that $x_n y_n \rightarrow 0$.

My idea was that, since $x_n \rightarrow 0$, multiplying some number of $y_n$ by $0$ would always just be $0$, and we know that we can do this since $y_n$ is bounded (although I don't think it would matter if it were unbounded).

Am I right in thinking this, or is the answer in a completely different direction?

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    To understand the importance of having $y_n$ bounded you might try your method with the sequences $x_n = \frac1n$ and $y_n=n$.2012-10-19

3 Answers 3