Instructions Provide a self-explanation of the proof of the following theo- rem. Theorem Every bounded, increasing sequence is convergent. Proof Suppose (an) is a bounded, increasing sequence and let A := {an: n E N}. Then A has a least upper bound and let a := sup(A). We claim that (an) converges to a. Let e > 0. There exists an E A such that a -e < an. It follows that an < an for all n > N. Therefore a- € < an N. Hence lan – al < e for all n > N. We have shown that (an) converges.