How to write a personal message on behalf of a telegram bot to a specific user?

Question

Python

I know the username of the user, I know that the person should already be running the bot. Just after launching the bot, how to write a private message to a specific user on behalf of the bot? And how to write to everyone on behalf of the bot (who ran it before).
In addition to the code, you can also explain how to do this to the user in the telegram bot itself, please.