Are there any proven methods for increasing bone density and bone strength?

Yes, there are. 

In children before puberty, participation in high impact sports like gymnastics has shown that they have stronger thicker bones through adulthood [1].

In adults, progressive resistance training (continuously overloading your body with resistance, also known as weight training) has demonstrated in over 24 clinical studies to improve bone density. [2] [3]

Some of these studies directly targeted elderly people, some targeted younger people.  Most of them found increased density in most of the skeletal areas.

Running can also increase bone density because of the impacts on the ground. [4] The big caveat is you must eat enough. Studies have found that young female distance runners have height growth issues and bone density problems because they either don't have the time or are unable to eat enough to rebuild their muscular-skeletal system after their workouts. [5]

It is essential to eat properly while training. Be sure to get enough Calcium and vitamins to ensure the bones have enough material to build themselves up. Calcium is found in dairy products and leafy greens. Vitamin D is essential in protecting your bones and ensuring that calcium can be absorbed. You can get vitamin D from sun exposure, fatty fish, and some dairy products have vitamin D added, along with some orange juices.  Without Vitamin D, you can't get enough calcium.  [6]

Try to get both calcium and vitamin D from food and the environment before resorting to supplementation.

[1] Female Gymnasts: does their training reduce growth rates, delay maturation and increase the risk of long-term skeletal injury?

[2] The effects of progressive resistance training on bone density: a review.

[3] Resistance Training and Bone Mass

[4] Impact of running on lumbar bone density: a 5-year longitudinal study.

[5] Long-Distance Runners Risk Bone Loss

[6] Calcium and Vitamin D: What You Need to Know

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