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

First, assume that (Sn) is bounded. This means that there exists some M > 0 such that |Sn| ≤ M for all n ∈ ℕ. We want to show that lim sup Sn < +∞. By definition, lim sup Sn = limn→∞ sup{k ≥ n} Sk. Since (Sn) is bounded, we know that sup{k ≥ n} Sk ≤ M for all n ∈ ℕ. Therefore, lim sup Sn ≤ M < +∞, which proves the desired inequality.

Conversely, assume that lim sup Sn < +∞. We want to show that (Sn) is bounded. By definition, lim sup Sn = limn→∞ sup{k ≥ n} Sk. This means that for any ε > 0, there exists some N ∈ ℕ such that sup{k ≥ n} Sk ≤ lim sup Sn + ε for all n ≥ N. Let ε = 1, then there exists some N ∈ ℕ such that sup{k ≥ n} Sk ≤ lim sup Sn + 1 for all n ≥ N. In particular, sup{k ≥ N} Sk ≤ lim sup Sn + 1. Since sup{k ≥ N} Sk is finite, we can define M = max{|S1|, |S2|, ..., |SN−1|, |sup{k ≥ N} Sk| + lim sup Sn + 1}. Then, we have |Sn| ≤ M for all n ∈ ℕ. Therefore, (Sn) is bounded.

Overall, we have shown that (Sn) is bounded if and only if lim sup Sn < +∞