Lifting weight is not the only answer to building upper body strength. Using your body weight as a form of resistance is a great alternative to weight training. You just need to find the right exercises to effectively challenge your upper body muscles and look to implement progressive overload into your training to keep up your strength gains.
To help you get started, here's a five move routine from fitness trainer, Joey Bronston. Since it's a fuss free no equipment workout, the main thing you'll need to consider before jumping into the upper body exercises is space and comfort.
