Let (an) be a bounded sequence.Recall the definitions lim inf a, = lim,→(inf(an)) and lim sup a, = lim,→ (sup(a,)). Show that: 1. sup(an) is a proper sequence and is

monotone decreasing. 2. Show that sup(an) converges. 3. Let S E R. Show that S = lim sup an if and only if: for every ɛ > 0 there are infinitely many n E N such that an > S – ɛ, and there are only finitely many n such that an > S + ɛ. 4. Conclude that if S = lim sup an, then for every ɛ > 0, almost always an < S + ɛ .

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