We went from late 20s temperatures to -9, ouch, and neither of us had a coat! It was just too cold to do too much. We did go and see Tiananmen Square, one of the largest city squares in the world:
And the Forbidden City, where the emperors lived exclusively for 500 years from 1400.
(I know this is naughty, but here is a photo from Wiki that shows the magnitude of the site as viewed from Jingshan gardens to the north, we were staying in a Hutong just next to these gardens):
We managed to squeeze in as much local food as possible! We also went to watch an acrobatic show one night which included a death defying motorbike stunt in a metal globe, google "Chinese acrobats motorbike stunt" if you want to see it on youtube!