Solipsism and its problems.

Solipsism is the philosophical idea that only one's own mind is certain to exist. Solipsism is an epistemological or ontological position that knowledge of anything outside one's own specific mind is unjustified. The external world and other minds cannot be known and might not exist. In the history of philosophy, solipsism has served as a skeptical hypothesis.

Problem One: Explaining Solipsism to someone
However a problem is faced in Solipsism when you try to explain this belief to someone else. You see, it is nearly impossible to convince anyone that this could be true. There are only two ways to convince someone that Solipsism is true:

You have to convince them that *they* are valid, and that you yourself are a projection of their own self. However they'll know you can't possibly believe this yourself.

So, you have to convince them that they are a projection of your imagination. Which they simply will never believe, because they are conscious themselves.
It appears, using this simplistic approach to Solipsism, to be impossible to state that you yourself are the only valid consciousness in a meaningful way, because anyone who you tell it to will automatically disbelieve you. That is a major problem, it makes Solipsism workable only as a completely private belief. It doesn't necessarily mean it isn't true, if you do honestly believe that all the people around you are automatons of your own creation, and do not possess independent consciousness.

Problem Two: There can only be one!
It seems only one person in the world can be a solipsist and be correct in their belief. If two Solipsists ever meet, one of them are wrong in their belief. This problem begins to appear as a symptom of a great problem of solipsism: It might not actually be true.

Can somebody refute my problems?