Proof of Integer Parts of Real numbers

1. Jul 25, 2010

TaliskerBA

I am struggling to understand the proof for integer parts of real numbers. I have used to mean less than or equal to because I could not work out how to type it in. I need to show that:

∃ unique n ∈ Z s.t. nx<n+1

The proof given is the following:

Let

A={k∈Z : kx}

This is a non-empty subset of R that is bounded above. Let α = sup A. There is an n ∈ A such that n > a - 1/2. n>α−1. Then nx and,since n+1>α+1>α, n+1̸∈A. Hence,n+1>x.

In particular I don't understand how A is bounded above, because I thought A = [k,∞) which has no upper bound. Where have I gone wrong?

2. Jul 25, 2010

mathman

A consists of all integers < x, so it is bounded from above.