Question

Prove or disprove there is a smallest positive number

Answer 1

\(n<n+1\\ \infty<\infty+1\\ \infty<\infty \\but ~~ \infty=\infty\)

Answer 2

You can't specific the smallest positive number. If you can think of extremely small positive number, there is always smaller positive number.

Answer 3

Assume that number is in real number set

Answer 4

the way i see it is that there is no such thing as a smallest positive number.
if i let x>0 I assume that for all numbers y>0. It is true that y>x
so if that is true then that means 1/2x>x

divide my x
1/2>1

Answer 5

\[\text{ Assume } x \text{ is the smallest positive number.} \\
\text{ Let } x>\epsilon >0 \]
\[\text{ So } x-\epsilon<x \]
\text{ and } x-\epsilon is also a what kinda number?

Answer 6

\[\text{ Assume } x \text{ is the smallest positive number.} \\
\text{ Let } x>\epsilon >0
\text{ So } x-\epsilon<x

\text{ and } x-\epsilon \text{ is also a what kinda number? } \]

Answer 7

that kind of forces\(\large \epsilon \) to be the smallest positive number right

Answer 8

I don't know, doesn't sound right. you assumed that x is smallest positive number, yet you set \(\epsilon \) to be number between \(x\) and \(0\), so that contradicts what you assumed before

I'm new to proof, so I dunno lol.

Answer 9

oh true

Answer 10

we may try below :
\(\large \dfrac{x}{2} \lt x\)

but the real big deal is in cooking p a nice argument around it

Answer 11

http://en.wikipedia.org/wiki/Proof_by_contradiction#No_least_positive_rational_number

Answer 12

That's kind of what we are tying to do. figuring out how to prove it using proof by contradiction.

Answer 13

\[\large \forall x \in \mathbb{R^+}\exists y\in \mathbb{R^+} : y \lt x\]
we define \(\large y = \dfrac{x}{2}\)

Answer 14

don't bother about it, im just brushing up my knowledge in quantifiers.. ;p

Answer 15

@nerdguy2535 is really good at proof writing

Answer 16

By number do we mean integer? or rational/real?

Answer 17

@helpmath123

Answer 18

Looking at what people have been typing, I assume its either rational or real.

Answer 19

lol he is offline...

Answer 20

Then @ganeshie8 's idea is the correct one. For any positive real number \(x\), \(\frac{x}{2}\) is another positive real number which will be strictly smaller.

Answer 21

What about rational? Should proof be different?

I think same proof can be used for rational, since this set is uncountable infinite set, like real number, right?

Answer 22

Yeah, the same proof works for rationals, since \(x\in \mathbb{Q}\Longrightarrow \frac{x}{2}\in \mathbb Q\).

Answer 23

Okay cool.

Answer 24

ahh good catch, didn't notice it was not mentioned whether the number was rational or real lol