Sum with Maximum is Maximum of Sum

Theorem
Let $a, b, c \in \R$ be real numbers.

Then:
 * $a + \max \left\{ {b, c}\right\} = \max \left\{ {a + b, a + c}\right\}$

Proof
, there are two cases to consider:


 * $(1): \quad b \ge c$
 * $(2): \quad b < c$

First let $b \ge c$.

We have:

Then:

Now let $b < c$.

We have:

Then:

Thus the result holds in both cases.

Hence the result, by Proof by Cases.