Hi, im trying to understand the following proof: If f is continuous on [a,b] and f(a) < 0 < f(b), then there is some number x in [a,b] such that f(x) = 0.(adsbygoogle = window.adsbygoogle || []).push({});

It starts with defining A to be {x: a <= x <= b and f is negative on [a,x]}. It uses the least upper bound property to determine that a least upper bound for A does exist and its between a and b. Call this least upper bound "u". Now suppose f(u) < 0. By a continuity theorem there is a d > 0 st f(x) < 0 for u-d < x < u+d. Now there is some x0 in A which satisfies u-d < x0 < u.

This is where i dont understand. Intuitively it makes sense, but how do you know that theres any x between u-d and u+d or u-d and u. All i know is that any x0 in A must be smaller or equal to u. Im trying to make as little assumptions as possible and it seems as if this statement assumes the intermediate value theorem, and thats basically what its trying to prove.

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# Homework Help: Help with proof

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