@eyinyangb So, this entirely depends on the day to day weather, because Chicago weather is absolutely insane and bipolar. You completely can, but you probably won't know until you get here and actually experience the winter weather. Everyone reacts differently to the weather (3 weeks ago, almost everyone was back in decent weight jackets, I wore a tee shirt. It's subjective.)
A mile is not too far. I just got done with my undergrad in Chicago, I lived about a mile from my closest school building, I've walked that before. If you have the proper winter gear, you can. It's not advisable sometimes, like when it's 10 degrees out plus whatever the wind chill is, or when it's snowing. See if UChicago will provide a Ventra UPass with your tuition. You'll get unlimited bus and L rides for the semester, so it's very easy to justify a bus ride to campus when it's nasty out. Plus, you know, invest in a down parka, a ton of scarves, hats, and good gloves, and nice sturdy boots.
On the flip side, we had a really mild winter this year (as in, the temperature hovered right around freezing with a few significant drops, but nothing too bad). And if the wind isn't blowing, the cold isn't that bad.