Karl Marx and Class Mobility

So I've been reading Karl Marx recently, he is the father of communism. His basic theory is battle of the classes - that the capitalists oppress the proletariat - or the working class. The capitalist employs the worker and becomes rich through the fact that the value of the labor is higher than what the worker is paid. And the worker is usually paid just enough to live on. The situation as he describes it is, for the most part, correct. The thing that he fails to address, however, is that the proletariat HAS THE ABILITY to become a capitalist. There is class mobility. Yes, some capitalists have money that is inherited. But that just means that their relatives were smart. That is not unfair in any way, to pass money onto your relatives. It's a bit of a conflict, since I do feel a lot of the working class are struggling, and in an ideal world, we should help them all, but the ideas that Marx proposed are not the solution.