You can definitely find places for $850 and under in nicer neighborhoods. A good friend of mine lived in Logan Square right off of Kedzie (not a bad area) and had a HUGE garden apartment for $850. Another lives in another part of Logan Square, has an apartment with a living room, dining room, kitchen, hallway, bathroom, bedroom, front door, back door - $775. Another lives in Wicker Park/Bucktown (smaller apartment but still suitable for one person), $750.
So, as long as you are willing to look around, you can find great places. I live out in the suburbs and I pay $880 for my one bedroom apartment....a lot of my friends have cheaper and bigger apartments than me and they live in the city. Keep in mind that it costs money to commute as well, whether it be on the metra train or the EL.
Studio Available 6/1 or 7/1! $850
Rehab 1 Bedroom Logan Square apartment
there are two...amongst the many that you will find on craigslist and beyond. trust me, live in chicago.