Let (an) be a bounded sequence. (a) Prove that the sequence defined by yn = sup{ak k ≥n} converges. (b) The limit superior of (an), or lim sup an, is defined by lim sup an = lim yn where yn is the sequence from part (a) of this exercise. Provide a reason- able definition for lim inf an and briefly explain why it always exists for any bounded sequence. (c) Prove that lim inf an ≤ lim sup an for every bounded sequence, and give an example of a sequence for which the inequality is strict.

Suppose {xn} is a bounded sequence. Define bn where lim inf xn :_(n→∞) = lim inf bn :_(n→∞). Show that {bn} is an increasing sequence.

Show that if (XN) is bounded sequences, then (XN) Converges if and only if lim sup (XN) = lim iNf (XN)

limit of the sequence lim h→∞ （1+2/h)^h