the North,the South,the East, the West are proper noun.

if they are proper noun then why some times written in small letters.


The sun rises in the East.

The stadium is in the north of the city.
The sun rises in the east (though you may sometimes find it miscapitalized).

The directional notations are normally capitalized only as a part of a proper name (North Korea, Bank of the West, East India Company), when referring to a specific area (the Confederate flag still flies in some parts of the South), in certain conceptual uses (in the East, courtesy is cultivated to a finer degree), and in a few cases, I will guess, by local practice. They are also capitalized on map compasses and similar locations.

As nouns of simple direction, they are common nouns (I live to the north of the city center; the main routes run from the east and the west).
The stadium is in the north