To prove: If limn→∞|an|=0, then limn→∞ an=0 .

Theorem used: Squeeze Theorem

“If xn≤zn≤yn for n≥N and limn→∞xn=limn→∞yn=L , then the value of limn→∞zn is L.”

Proof:

Given that, limn→∞|an|=0 . (1)

−(limn→∞|an|)=0

limn→∞(−|an|)=0 (2)

To prove that, limn→∞ an=0 .

Since −|an|≤an≤|an| for all n and apply the Squeeze Theorem, the limit of the sequence is obtained as follows.

limn→∞(−|an|)≤limn→∞an≤limn→∞|an|

By equations (1) and (2), 0≤limn→∞an≤0 .

Hence, limn→∞an=0 is proved.