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The narrator takes a little time out from the action to explain just how it is that Becky and Rawdon are able to live their fancy-pants lifestyle despite being totally broke.
There's really only one keyword to solve that mystery – credit. Namely, unguaranteed, far-too-risky credit.
(Incidentally, you may remember this kind of credit from the economic meltdown that took out our country's financial system recently. Good times!)
So yes, basically Becky and Rawdon put on a fancy front and get credit extended to them just on the basis of them looking the part, rather than on any kind of actual collateral or income.
The narrator points out that in fact many people live this way, totally beyond their means.
To make a little ready cash, Rawdon is always happy to play some cards or dice or billiards. Since he practices every day, he is really good at gambling. He's even perfected the routine of faking being a bad player until the antes are high and then coming through in the end to win.
Becky flirts with all the men who come to their house to gamble, and this encourages more traffic.
But Becky isn't satisfied; she wants to be just a little more established. She's hoping to get Rawdon some kind of government appointment.
When Miss Crawley dies, Becky goes into deep mourning.
Faking the need to go deal with inheritance issues, the Crawleys leave Paris for Brussels. They leave all their bills unpaid, defrauding everyone from the hotel owner to the dressmaker to the servants to the nanny.
Oh, yes, nanny. Remember how Becky had a son? He is also named Rawdon, but we'll call him Rawdon Jr. Either way, Becky cannot be bothered to deal with him, though Rawdon seems to be quite attached to his son.
In Brussels, Becky leaves Rawdon Jr. with her maid (who watches him so well that he almost drowns), and goes to London.
There she deals with all of Rawdon's creditors basically by buying out his debts pennies on the pound. (Brain Snack: Rawdon cannot come to London because at this time debtors were arrested and put into debtors' prison. Because nothing makes people earn money like putting them in a place where they can't work.)
Becky explains to the loan sharks exactly how broke Rawdon is and how he's never going to get the money to repay them. If they don't take the little that she's offering, they won't ever get any money at all; Rawdon will just never come back to England. She's very convincing, and all of the lenders see reason and settle.