How to keep elderly relatives socially active without overwhelming them

Lately I've been thinking about how challenging it can be to keep an older relative socially engaged when they slowly stop going out as much. My uncle used to enjoy meeting friends and participating in small community events, but lately he rarely leaves the house, even though he insists everything is fine. When I invite him for a short walk or suggest a simple visit to someone he likes, he often refuses because he says he gets tired too quickly or doesn't want to be a burden. I’m worried that this isolation might affect his mood, but I also don’t want to pressure him. Has anyone found a gentle, respectful way to help an older person reconnect with social activities without making them uncomfortable?