Prove Sm is bounded if and only if lim sup Sn +00

First, assume that (Sm) is bounded. Then, there exists some M > 0 such that |Sm| ≤ M for all m ∈ N. Let Sn* = sup{Sm : m ≥ n}. Then, we have Sn* ≤ M for all n ∈ N, since the supremum of a set cannot exceed its upper bound. Therefore, lim sup Sn* ≤ M < +∞, which implies that lim sup Sn < +∞.

Conversely, assume that lim sup Sn < +∞. Then, there exists some M > 0 such that Sn* ≤ M for all n ∈ N, where Sn* = sup{Sm : m ≥ n}. Since Sn* is non-increasing as n increases, we have Sn* ≤ Sn+1* for all n ∈ N. Therefore, |Sm| ≤ Sn* ≤ M for all m ≥ n, which implies that (Sm) is bounded