Prove whether the statement is true or false. If the image of the continuous fun

Question

Prove whether the statement is true or false. If the image of the continuous function f:(0,1) -> R is bounded below, then the function has a minimum.

Explanation / Answer

proof Let f : D ? R be a function with domain D. The image of f is the subset of R de?ned by f(D) = {y ? R|y = f(x) for some x ? D}. We say that the function f attains a minimum value on D if there is some number m ? f(D) such that y ? m for all y ? f(D). In that case, m is said to be the minimum value and any number x ? D such that f(x) = m is said to be a minimizer. Example: The function f : (0, 1) ? R de?ned by the formula f(x) = 1/x does not achieve a maximum or minimum value. The image f(D) is not even bounded above and so there cannot be a maximum. It is bounded below as the values never get smaller than 1, but the minimum is not attained as it never even takes the value 1 on D = (0, 1). Assume there is no such number M. We will reach a contradiction and conclude that M must exist. De?ne a sequence an in D as follows. Let an be a point in D with the property that f(an) > n. (Such an an must exist, otherwise M = n would be an upperbound for the image and we are assuming none exists.) By Theorem 2.36 (Sequential Compactness), there is a subsequence {ank } that converges to some number a ? D. Since the function f is continuous at x = a, the sequence {f(ank )} converges to f(a). But, a convergent sequence is boundedwhich contradicts the fact that f(xnk ) > nk ? k for all k.

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