I believe we are dealing here with the Supremum of a function, that is, the LEAST UPPER BOUND.
This is absolutely trivial but I think the OP is a newbie in the subject. So let's start from the beginning.
It is correct that the range (the image of the domain D of f, which you denote as D) is a bounded set, by the definition of bounded function.
According to a crucial property of the real numbers (which?), any bounded subset of the reals has a supremum.
You can also show, using the definition of supremum, that the supremum is always UNIQUE, that is, there can only be one supremum for any given set, if there is any at all. There are subsets of the reals numbers (including the whole real line itself, considered as a subset of itself) that do NOT have an upper bound, let alone a least upper bound. The aforementioned property of the reals numbers is quite important because it serves to distinguish the reals from the rational numbers, for example.
Since f and g have each a UNIQUE supremum, it makes sense to define <br />
U \left (f \right) = \sup_{x\in D}\{f(x)\}= \sup\{f(x) : x\in D\}<br />
for f and g.
Note that if you "define" something you have to prove that it is WELL-DEFINED, that is, that it exists, that it doesn't depend on which way you represent the initial object you will associate it with (in this case the function f, which no matter if you write it as f-1 +1 or 2f-f, it still has the same supremum...this is something obvious in this case but in other areas of mathematics you will work with equivalence classes and this will not be so obvious) and that it is UNIQUE.Now, recall what a supremum is: as its name states it is THE SMALLEST UPPER BOUND OF ALL for that given set. Therefore, if you find any upper bound of that set, it must be true that this is bigger or at worst equal to the supremum.
From the mere fact that a supremum is, by definition, an upper bound we get
<br />
f(x) \leq U(f)
and also that
<br />
g(x) \leq U(g)
This implies that
<br />
(f+g)(x) \leq U(f) + U(g)<br />
for all x\in D. This means that U(f) + U(g) is an upper bound for the set (f+g)(D). What does this imply according to the definition of the supremum of this last set?
For the second part, you need to prove that the the supremum of the sum is NOT the sum of supremums. This means that ocasionally we can get the equality to hold but in GENERAL this won't happen. Take the functions f(x)=x and g(x)=1-x defined in the interval [0, 1] and therefore, bounded in that domain. Check that the equality does NOT hold. This single COUNTEREXAMPLE (in which we specialize in a particular case using allowed functions and a specific interval) proves the assertion in the second part of your problem.
Since you only need ONE counterexample to show that something is not true, This part of the problem can be rephrased as saying "Show that the equality cannot hold identically". (as opposed to "ocassionally" because it CAN hold sometimes).
