# 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.

Related Precalculus Mathematics Homework Help News on Phys.org
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
Homework Helper
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?