Question

Prove that there is no smallest positive real number?

Answer 1

I can divide your number by 2 and get a smaller one.

Answer 2

lets say your number is a.
divide it by a positive number b.
a(1/b)
keep doing it
a(1/b)(1/b)

a(1/b)^n will never hit 0 but will keep becoming smaller with every increase in the number of times you divide it.

Answer 3

This the same argument as proving there is no largest positive real number.

Answer 4

By archimedes principle: given a<b there is some n in natural s.t. na>b, and given a>0 there is some n in natural s.t. 0 < 1/n < a

Answer 5

explain it little bit more plz....

Answer 6

when proving something you need to show theroems/lemmas/remarks about what you claim....

Answer 7

do you know what a natural number is?

Answer 8

ya

Answer 9

1, 2, 3, 4, ...

Answer 10

so you give me any number like: .00000000000000000000001

there is some 1/n that is smaller like 1/10000000000000000000000000000000

Answer 11

you could talk about density also, given and real number a<b there is some real number c such that a < c < b
let a = 0

Answer 12

What if a=0.999... and b=1 ?

Answer 13

add a 9

Answer 14

between any two numbers there is an infinite amount of numbers

Answer 15

O.9999...

Answer 16

you actually did not show that you added another 9. it would be at the end of your ...

Answer 17

:)

Answer 18

There is no end to an endless string of 9s.

Answer 19

exactly

Answer 20

thus there is always a smaller number

Answer 21

or a number inbetween I should say

Answer 22

So there is a number in-between 1 and 0.999...?

Answer 23

.9999........ with endless 9's is not a finite number, just like infinity is not a finite number.

Answer 24

so you are asking a question that does not make since

Answer 25

1/3 =0.333... is a finite number!

Answer 26

1/3 is and the aproximation .33333333333 is an aproximation of 1/3 and yes its finite, but .333333333...... is not a real number if there are infinite 3's

Answer 27

we would say it approaches 1/3

Answer 28

2/3=0.6666...

Answer 29

1/3+2/3=1 the last time I checked

Answer 30

same thing, if you cant write the number its not finite, and you cant write .666666........(notice you have to do ......)

Answer 31

yes, but .333333333+ .6666666666 = .9999999999 != 1

Answer 32

you can write .333 untill the universe implodes, it will never be 1/3

Answer 33

http://en.wikipedia.org/wiki/0.999...

Answer 34

lol, you are not getting my point. Can you write .333333......?

Answer 35

0.999...=1

Answer 36

No it doesn't.

Answer 37

it approaches 1, and is not finite untill you stop writing the 9

Answer 38

There is no "smallest" number in-between 0.999... and 1.