Set Addition with the Reals, The Suprema Lemma

We'll need some properties of $Q, N$ to be able to determine this suprema. So let's get to those:

c + A

Let $A \subseteq R$ be nonempty and bounded above, and let $c \in R$ . Define the set $c + A$ by:

c + A = {c + a : a \in A}

Then $sup (c + A) = c + sup (A)$ .

Proof
For (i), set $s = sup (A)$ . We see that $a \leq s$ for all $a \in A$ consequently, but then $a + c \leq c + s = c + sup (A)$ as required.

For (ii), let $b$ be an arbitrary upper bound for $c + A$ , so then $c + a \leq b$ for all $a \in A$ . Then $a \leq b - c$ for all $a \in A$ . So then $b - c$ is an upper bound for $A$ . Since $s$ is the least upper bound for $A$ , then we have it that $b - c \leq s$ , thus $b \leq c + s = c + sup (A)$ , completing (ii).

Therefore, $sup (c + A) = c + sup (A)$ .
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