First, I think OP should edit the eradicate in his title pose to either ameliorate/curtail or reduced, give it a soft landing because it can not be eradicated in any society. And, if I have to make inference from the bible, do we still remember the rich man and the poor man story?
There are a lot of wrong statements in your post. A lot of wealthy people worked hard for that wealth, and no luck was involved.
I think that there is also the place of luck in our lives and being rich. The difference is that, being ready to identify the luck; in which case, the rich could have been ready. Off course, when being ready/opportunity meets luck, it takes one to wealth.
So we have to live ready.
I think of the united state as a country which has evolved a lot over the decades. It was initially a colony of people with pathetic financial statements and outdated infrastructure. The country paid huge attention to the human development and the current studies about their economy and political structure conclude them the most powerful country of the world.
So i want to say that they have raised their per capital income immensely and in a sense have put a full stop to the poverty. The word eradicate represent an optimistic approach towards finishing poverty but again what maters the most is the real figures and this can be done so. Poverty eradication or amelioration is a long journey and is the challenge being posed to so many countries currently.