Yes, it is grammatically correct. However, it also sounds very poetic. :) Here, you are making a remark about your status of being, and reflects deep thought about your inner self. It is more natural in conversation to instead say, "I am becoming more patient and persistent through playing this game." This sounds more like a simple remark rather than a deep philosophical change within yourself. The difference is very subtle... I'm sorry if I didn't explain it well!