For (a) you have the right idea, but it would be much more convincing if you simply gave a concrete example of a random variable which is not bounded.
Q: Let X be a random variable.
a) Is it necessarily true that there is some real number c such that X + c >= 0?
b) Is the above true if the sample space is finite?
Approach:
I believe that a) is false and b) is true. The question is basically asking whether or not we can find a bound on X. That is to say is there a c where X(z)>=c for all z in the sample space.
For a), my idea was to state that since X is infinite, there is not necessarily a minimum value to X. Therefore it becomes impossible to find a c such that c<=X(z) for all z because c-1 might belong to X but c-1<c.
For b), my idea was to take any finite set and state that because the set is finite, there must be a minimum value to X. Let us assume this minimum value occurs at X(z). If we let c=X(z), then it becomes true that for w =/= z, X(w)>=X(z)>=c.
Any thoughts or suggestions would be greatly appreciated.
Since you only commented on a) I'm assuming that I had the correct idea for b). Either way, thanks for the suggestion. You made me realize that it only takes providing one example to contradict the statement, which is much easier than going through a (perhaps) rigorous proof. Thanks again.