A short sale offer just a week before foreclosure really isn't enough time. I'm sure someone will come forward with a story about how they did a short sale in 5 days, or whatever, but you really have to allow a couple of months. And sometimes that isn't enough.
Having said that--and in the case of making a short sale offer just a week before foreclosure, there wasn't enough time--sometimes lenders still prefer to foreclose. Some may feel that they can get more for the property--or lose less--by selling it themselves than by accepting a short sale offer. Others sometimes feel that property values will rebound strongly, and they'd rather sit on a property for 6-12 months, if they can get, say, $100,000 more for it later. Others sometimes look at the equity in the property and decide to go for that. I spoke to an owner being foreclosed on earlier this week. He'd put $100,000 down on a home a number of years ago...paid down the mortgage some...value of the property had gone up. But, as a small business owner, ran into a cash crunch and missed one or two mortgage payments. He wanted to work it out with the lender, but the lender refused. So he's probably going to lose his home.
Those are just a few reasons why a lender would foreclose.