how to prove my christian friend that he can work on the sabbath day?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

It is better, dear brother questioner, to focus on convincing that person (whom you described as your friend) that trinity, which he believes in, is false and contradicts what Prophet 'Eesaa (Jesus) called for. Calling him to Islam should come in priority to convincing him that working on Sundays is permissible.

A Muslim must be aware of the priorities of Da'wah (calling to Islam) and must not disregard what is important and focus upon what is less important.

Kindly refer to fataawa 91563, 48349 and 88293. The last Fatwa is about the Muslim/non-Muslim relation.

Allaah knows best.