Gymnastics requires a great deal of upper body strength. While weightlifting is a valid option for building strength in your upper body, a better idea for gymnasts is to use body-weight exercises. When you use your own body weight for strength training, you not only build strength, you also improve your balance. Start with three sets of 12 repetitions. And take one day of rest in between workouts.
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Lie face down on the floor with your palms flat on the floor just wider than your shoulders. Push up onto the balls of your feet, straightening your arms completely. Pull your abdominal muscles in tight toward your spine. Keep your body completely straight from head to heels, as you lower your body toward the floor, bending your arms to a 90-degree angle. Push your body back up, straightening your arms.
Use an exercise bench for this body-weight exercise. Sit on the front edge of the bench with your hands close to your rear and your knees bent to a 90-degree angle. The heels of your hands should be on the front edge of the bench, with your fingers pointing down to the floor. Lift your rear off the bench, and push slightly away from it. Bend your arms to a 90-degree angle, lowering your rear to just below the height of the bench. Push back up to straight.
Kick up into a handstand against a wall, pulling your abdominal muscles tight in toward your spine. Bend your arms, lowering your head toward the floor. Be careful not to bend so far that your arms give out. Push back up until your arms are completely straight. Once you have built enough strength and balance, perform one set away from the wall, in a freestanding handstand. Work up to performing all of your pushups without the aid of the wall. The first time you do a handstand pushup and until you are proficient at it, have a friend or trainer spot you.
Use the uneven bars or the parallel bars for this exercise. Grab the bar with your hands shoulder-width apart and the backs of your hands facing away from you. Dangle with your arms completely straight. Bend your arms, pulling yourself up until your chin is higher than the bar. Slowly lower your body until your arms are straight. Do not allow your feet to touch the floor. Your arms must lift all your body weight. If necessary, keep your legs bent to keep your feet off the floor.