I am working on a formal proof of the upper bound property of non-empty subsets of N:
Every non-empty subset of N that is bounded from above has within it, the least upper bound of that set.
I can't seem to get anywhere. Here are my thoughts so far, such as they are:
Let x be a non-empty subset of N. Let b be an upper bound of x. Suppose to the contrary that for every element of x, there is a still larger element in x. I should be able to obtain a contradiction from this, but how? Should I consider another approach? Any help would be appreciated.