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## Homework Statement

If S

_{1}, S

_{2}are nonempty subsets of ℝ that are bounded from above, prove that

l.u.b. {x+y : x \in S

_{1}, y \in S

_{2}} = l.u.b. S

_{1}+ l.u.b. S

_{2}

## Homework Equations

Least Upper Bound Property

## The Attempt at a Solution

Using the least upper bound property, let us suppose that a is an upper bound for S

_{1}and b is an upper bound for S

_{2}, as they are both a set of real numbers. Then there exists an x in S

_{1}s.t. x≤a and there exists a y in S

_{2}s.t. y≤b. By adding them together, x+y≤a+b where a and b are the upper bounds for their respected set of real numbers. Thus, it is equal to l.u.b. S

_{1}+ l.u.b. S

_{2}

I think this is how it goes. I'm sorry if my proof is horrible; it has been a significant period of time since I've done proofs, as statistics and probability classes don't use proofs! Also, I can't get \in to work. Sorry about that!