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inductive proof of binomial theorem

number theory
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Mathematics Subject Classification

05A10 no label found


I wanted some hints on this problem. Let f be a bounded function on interval I, prove that
sup({-f(x): x in I}) = -inf({f(x): x in I}).

We had a problem similar to this in our honors calculus class last year; The way I found best was to start by contradiction, that they are not equal.

That is, if f is a function f: I -> A, then there is some element a in A such that a < -inf({f(x) : x in I}) with a an upper bound for {-f(x): x in I}.

What then, can you say about -a and its relation to {f(x) : x in I}

Well, at least something like that.

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