Science:Math Exam Resources/Courses/MATH103/April 2014/Question 01 (e) i/Solution 1

If we expect that the sequence converges to zero then we can ignore the terms that already are zero. The remaining terms (let's call them $b_{n}$) are

$b_{n}=\{1,1/2,1/3,1/4,1/5,\dots \}=\{1/n\}.$

To determine if the sequence converges we compute

$\lim _{n\to \infty }b_{n}=\lim _{n\to \infty }{\frac {1}{n}}=0$

and indeed we see the sequence converges to the value zero. Since all the remaining terms in $a_{n}$ are already exactly zero then $a_{n}$ must converge to zero as well.

The statement is true.