Proving upper bounds? This is a relatively simple question.  Say you are given a set that exists in the reals.  How would you go about formulating a proof to prove that some number that is not given is in fact an upper bound?

One way to do it is this. By observation, one makes a guess what an upper bound would be. Then by using algebraic maipulations and properties, one proves that it indeed is an upper bound.
For example, consider the set $\{1-\frac1n:n=1,2,3,\ldots\}$. Visualising this set on the real line leads us to conjecture that $1$ is an upper bound. One then verifies that for any positive integer $n$, $1-\frac1n<1$ holds.
