Question

Suppose that the limit as n approaches infinity of Xn=0. If {Yn} is a bounded sequence, prove that the limit as n approaches infinity of XnYn=0

Answer 1

Task 1.

Suppose that the limit as $n$ approaches infinity of $x_{n}$ equals . If $\{y_{n}\}$ is a bounded sequence, prove that the limit as $n$ approaches infinity of $x_{n}y_{n}$ equals .

Solution.

Let $\varepsilon>0$ be a positive real number. Find $N\in\mathbb{N}$ such that $|x_{n}y_{n}|<\varepsilon$ for all $n>N$.

Indeed, since $y_{n}$ is bounded, there is $M\geq 0$ such that $|y_{n}|\leq M$ for all $n\in\mathbb{N}$. Furthermore, $x_{n}\to 0$, therefore, there is $N\in\mathbb{N}$ such that $|x_{n}|<\frac{\varepsilon}{M}$ for all $n>N$. Then

$|x_{n}y_{n}|=|x_{n}|\cdot|y_{n}|<\frac{\varepsilon}{M}\cdot M=\varepsilon$

for all $n>N$. Since $\varepsilon$ was an arbitrary positive number, the last means that $x_{n}y_{n}\to 0$ as $n\to\infty$. ∎

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