Prove using a proof by contradiction: There is no smallest positive real number

Question

Let us assume the contradiction: There is a smallest positive real number.

How do I continue?

You continue by constructing a smaller positive real number from it. — Daniel Fischer, Oct 05 '14 at 21:44
Can you please elaborate? I'm new to proofs and I've got a midterm tomorrow — user178415, Oct 05 '14 at 21:45
Given $x > 0$, you probably can think of hundreds of ways to obtain an $y$ with $0 < y < x$ from $x$ within minutes, can't you? — Daniel Fischer, Oct 05 '14 at 21:47
So can I just say: Assume there exists a smallest positive real number,x and x/2 is also real but its smaller. Therefore its a contradiction — user178415, Oct 05 '14 at 21:48
Just here to point out that "contraction" isn't the word to use here. Should say "Let us assume the contradiction" although I recommend "we proceed toward contradiction" or "For the sake of contradiction, suppose..." — graydad, Oct 05 '14 at 21:51

score 2 · Answer 1 · answered Oct 05 '14 at 21:45

2

Let's suppose there is a smallest positive number, say $s$. So what is $\frac{s}{2}?$

answered Oct 05 '14 at 21:45

graydad

1 Answers1