Show that for every n, n<=x<n+1

[joedirt658]

Homework Statement

Assume that for every positive real number x there is an integer n>x.
Show that for ever real number x there is an integer n such that n<=x<n+1.

Homework Equations

none

The Attempt at a Solution

I think i am supposed to use the well ordered principle of induction to solve this problem so here is my solution.

Suppose there are two sets S and K and when combined they form the set of all integers.
Let K contain everything not in S
if K contains every integer greater than x and S contains the numbers {-∞,..., n-2, n-1, n} then the smallest integer in K is n+1 and therefore
x<n+1

if S contains everything that is not in K, then the integers in S can either be less than or equal to x because both situations satisfy the condition that the integer is not greater than x.

therefore n<=x.
by the transitivity property we can say that
n<n+1 and n<=x<n+1.

i am not sure if i did this correctly and since i am self teaching myself some things in math that i feel i do not have a good basis in, it would be cool if you guys could check my answer.

thank you in advance

 

[ptolema]

Are you trying to do induction on natural numbers or real numbers? As far as I know, induction isn't used much to prove something for all real numbers, just for all natural numbers.

 

[HallsofIvy]

Let S be the set of all positive integers greater than x. By the hypothesis, that set is non-empty. Can you show that S contains a smallest member? If so, how does that give you your proof?