Science:Math Exam Resources/Courses/MATH220/April 2005/Question 10/Solution 1

Since $f$ is bounded there exists a constant M such that

$|f(n)|<M\quad {\text{for all }}n\in \mathbb {N}$

We will show that $\{f(n)a_{n}\}$ converges to zero. Let $\epsilon >0$. Since $\{a_{n}\}$ converges to zero there exists a positive integer N_ε such that

$|a_{n}|<{\frac {\epsilon }{M}}\quad {\text{for all }}n>N_{\epsilon }$

Hence

$|f(n)a_{n}|=|f(n)||a_{n}|<M{\frac {\epsilon }{M}}=\epsilon \quad {\text{for all }}n>N_{\epsilon }$

This shows that $\{f(n)a_{n}\}$ converges to zero.