# inductive proof of binomial theorem

## Primary tabs

Keywords:
number theory
Major Section:
Reference
Type of Math Object:
Proof

## Mathematics Subject Classification

05A10 Factorials, binomial coefficients, combinatorial functions

### Sups and infs

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}).

### Re: Sups and infs

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.