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February 21st, 2013, 06:44 PM  #1 
Member Joined: Oct 2011 Posts: 31 Thanks: 0  intermediate value theorem
For the following proof of the intermediate value theorem ,which i found in wikipedia: Proof: Let S be the set of all x in [a, b] such that f(x) ? u. Then S is nonempty since a is an element of S, and S is bounded above by b. Hence, by completeness, the supremum c = sup S exists. That is, c is the lowest number that is greater than or equal to every member of S. We claim that f(c) = u. Suppose first that f(c) > u, then f(c) ? u > 0. Since f is continuous, there is a ? > 0 such that ?f(x) ? f(c)? < ? whenever ?x ? c? < ?. Pick ? = f(c) ? u, then ?f(x) ? f(c)? < f(c) ? u. But then, f(x) > f(c) ? (f(c) ? u) = u whenever ?x ? c? < ? (that is, f(x) > u for x in (c ? ?, c + ?)). This requires that c ? ? be an upper bound for S (since no point in the interval (c ? ?, c] for which f > u, can be contained in S, and c was defined as the least upper bound for S), an upper bound less than c. The contradiction nullifies this paragraph's opening assumption. Suppose instead that f(c) < u. Again, by continuity, there is a ? > 0 such that ?f(x) ? f(c)? < u ? f(c) whenever ?x ? c? < ?. Then f(x) < f(c) + (u ? f(c)) = u for x in (c ? ?, c + ?). Since x=c + ?/2 is contained in (c ? ?, c + ?), it also satisfies f(x) < u, so it must be contained in S. However, it also exceeds the least upper bound c of S. The contradiction nullifies this paragraph's opening assumption, as well. We deduce that f(c) = u as stated. I have the folowing questions: 1)How does the author come to the conclusion that ,c? is an upper bound for S ,in this part of the proof: "This requires that c ? ? be an upper bound for S (since no point in the interval (c ? ?, c] for which f > u, can be contained in S, and c was defined as the least upper bound for S), an upper bound less than c. The contradiction nullifies this paragraph's opening assumption" 2) How does the author conclude that ,x = c+?/2 is contained in S ,since he has not proved that in the following part of the proof: "Suppose instead that f(c) < u. Again, by continuity, there is a ? > 0 such that ?f(x) ? f(c)? < u ? f(c) whenever ?x ? c? < ?. Then f(x) < f(c) + (u ? f(c)) = u for x in (c ? ?, c + ?). Since x=c + ?/2 is contained in (c ? ?, c + ?), it also satisfies f(x) < u, so it must be contained in S" 3) Nowhere in the proof proves that : a<c<b 

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