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(1) Every convergent sequence contains either an increasing, or a decreasing subsequence.?
Proof: Suppose that lim(as n goes to infinity) x(sub n) = L. If infinitely many terms are equal to L, we are done. If not, either infinitely many terms are > L, or infinitely many terms are < L. Choose one option and complete the proof.(2) Adapt carefully the bisection argument seen in the proof of Bolzano-Weierstrass to prove the Intermediate Value
Theorem: If f is a continuous function on [a,b], and if f(a) and f(b) have opposite signs, then there exists a point c (a,b) such that f(c) = 0.
Proof: Suppose that f(a) < 0 and f(b) > 0. Denote a(sub 0) = a, b(sub 0) = b and consider the intervals [a, (a+b)/2] and [(a+b)/2, b].
Your boss calls you into his office and he is extremely angry. He was embarrassed at a meeting because he put forward a recommendation that you had made and his idea was shot down by one of the technical geniuses in another department.
Evaluate lim sup Ek
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