What is the Islamic ruling for saying to each other, when Ramadan comes,: "Ramadan Kareem" and "Ramadan Mubaarak". Is it a Bid'a or a Sunnah? And what is the proof of it? We never say this in my home country

Praise be to Allah, the Lord of the World; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Some scholars believe that it is lawful for Muslims to congratulate each other for the entering of Ramadan by saying "Ramadan Mubarak " and the like.
Their evidence for this is the Hadith reported by Ahmad that the Prophet said: "…. A blessed (Mubarak) month has shadowed you …".
Ibn Rajab said that this Hadith is an evidence for the lawfulness of congratulating one another for the coming of Ramadan'.
Allah knows best.