It happened in the 1200 block of East 54th Street on Sunday.
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Officers were responding to reports of mailbox theft when they encountered a man inside the building's entrance.
When approached by the officers, the suspect attempted to escape by jumping from a third-floor apartment.
The suspect was taken into custody and transported to University of Chicago Hospital in serious condition.
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Residents said the area has seen an increase in burglaries recently.
Police said charges are pending.
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