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I am given two sequences:

$\{a_n\}$ which converges towards 0 and $\{b_n\}$ which is a bounded sequence.

I have to show that $\{b_na_n\}$ is converging towards 0.

I seem to be quite stuck on how to approach this, since the rule i would like to use says that the two sequences has to both be converging.

I hope someone can help me out.

/Chris

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2 Answers

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Hint: Since $b_n$ is bounded then you know that $$|b_n|\le M$$ for some real number $M$ This excludes the case that $b_n \to \infty$ and therefore the case that the limit $a_n\cdot b_n$ is of the form $0\cdot \infty$ and thus indefinite. The above inequality implies that $$|a_nb_n|\le|M||a_n|$$ which gives you the result by letting $n \to \infty$.

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$a_n$ being a null sequence means that $\forall \varepsilon > 0$ there exists some value $N \in\mathbb{N}$ so that $$\forall n \geq N \ : \ |a_n| < \varepsilon \ .$$

$b_n$ being bounded on the other hand means that there is some positive $M$ so that $\forall n$ $$|b_n| \leq M \ .$$

Now we get for the product $|a_n b_n| = |a_n||b_n| \leq |a_n| M$ and also $|a_n| M < \varepsilon M$. Thus $$\forall n \geq N \ : \ |a_n b_n| < \varepsilon M$$

so define $\varepsilon' := \varepsilon M$ and you'll get

$$\forall n \geq N \ : \ |a_n b_n| < \varepsilon'$$

and rewrite the entire statement to

$$\forall \varepsilon' > 0 \ \exists N \in\mathbb{N} \ : \ \forall n \geq N \ : \ |a_n b_n| < \varepsilon'$$

which is the definition of $a_n b_n$ being a null sequence.

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