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

## 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.

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## 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.