Supremum of a Subset of Integers is an Integer

Question

An non empty subset of integers which is bounded above has a Supremum. How do we prove that Supremum is also an integer? Is my answer correct? :)

Let's take this set as A and Sup A =S.

Let's assume S is a Real number. (S is not an integer)

For all x in A; x<=S , S-1 < S

There exists n which is an element of A greater than S-1 , S-1 < n <= S

1. For all x in A; x<=n. n is an Upper Bound ; S<=n

2. But n is an element of A ; n<=S.

From 1. & 2. S<=n<=n ; n=S

n is an integer. So S is an integer. ; This is a contradiction.

So, our assumption is wrong. Therefore S is an integer.