TRUE-FALSE If your answer is \"True.\" justify by quoting a definition or theore

Question

TRUE-FALSE If your answer is "True." justify by quoting a definition or theorem, or by giving a proof. If your answer is "False." give a counter-example. Suppose S . the set of rational numbers. If S is bounded above, then S has a least upper bound (sup) s . If alpha is the supremum of the set S and k

Explanation / Answer

e) since Q is the set of rationals, this isnt true, s does not have to be in Q. Imagine the sequence (1+ 1/n)^n as n -> infinity. This sequence clearly is in Q (it can be written (n+1)^n/n^n), but its limit is e, which is not a rational number. Since it is strictly increasing and bounded above by e, the least upper bound is e, which is not in Q g) this is true, because the supremum of a set is the LEAST upper bound, so if k

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