Jungkook was a child who grew up without a parent so he met Taehyung's grandmother and she took care of him. Taehyung grew up with his father because his mother and brother left him and went to the US for the good life and found a new wife.

Taehyung's father died because he worked hard to bring back his former wife and child. The grandmother thought of making her grandson happy so she made a jungkook to write a letter to encourage him.

Papa! Someone gave me a love letter * excited * - taehyung

Really son what name - Papa

His name is jimin, grandma gave it to me-taehyung

Dad, can I write to him too-Taehyung

All right son as long as you're happy-Papa

And that's when taehyung and jimin started writing letters to each other but jungkook only got the name jimin in a newspaper so as not to know his real name