Proof: limsup|sn| = 0 if and only if limsn = 0

First, we prove that if limsn = 0, then limsup|sn| = 0.

Assume limsn = 0. Then, for any ε > 0, there exists an N such that for all n > N, |sn| < ε. Now, consider the sequence of non-negative terms |sn|:

|sn| = max{|s1|, |s2|, ..., |sn|}

For any n > N, |sn| ≤ ε. Thus, limsup|sn| ≤ ε. Since ε is arbitrary, we conclude that limsup|sn| = 0.

Next, we prove that if limsup|sn| = 0, then limsn = 0.

Assume limsup|sn| = 0. Then, for any ε > 0, there exists an N such that for all n > N, |sn| < ε. This implies that |sn| ≤ |sn| < ε for all n > N.

Thus, for any ε > 0, there exists an N such that for all n > N, |sn| < ε. This implies that limsn = 0.

Therefore, we have proved both directions of the statement, and we conclude that limsup|sn| = 0 if and only if limsn = 0.