It is not possible for one group of people to be rich unless another group of people is poor. This was pointed out by John Ruskin in 1860 in his series of essays Unto This Last. The original Unto This Last is difficult to read, but it was later translated by Gandhi into a paraphrase in Gujarati, which was later translated back into English under the title Unto This Last: A Paraphrase.
Ruskin describes wealth as being the power over the labour of others. This power is often symbolized by money. For instance, if I am rich and have lots of money, I can hire a poor person to clean my house. The poor person must do the work, as he needs my money to pay for food and shelter. The poor person can't hire me in return to clean his house, as, due to his lack of wealth, he does not have power over my labour because I don't need to bother getting money from him.
Ruskin further exposes that wealth inequality gets worse, because the wealthy can cause the poor to try to outbid each other and thus not get a fair wage. For instance, were I rich I could find out who would do the work for the least payment and hire only them. Outbidding each other like this, the poor could never get the wages needed to rise up the social ladder.
If you think there could be rich people without there being lots of poor people, just imagine that the state gave everyone 10 million dollars. No one could hire anyone to clean their house, because no one with millions of dollars would want to bother. The capitalist economy needs a reserve of poor people (preferably with many of them unemployed) to function, and thus the state, as long as the state is in the hands of capitalist interests, it will do little to address the roots of poverty and wealth inequality.