87. Prove Theorem 6. [Hint: Use either Definition 2 or the Squeeze Theorem.]
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.