prove that every subset of a finite set is a finite set?

Finite, no.

all finite set is countable.but,countable can be finite or infinite

A finite set has a finite number of elements, an infinite set has infinitely many.

In mathematics, a finite set is a set that has a finite number of elements. For example, (2,4,6,8,10) is a finite set with five elements. The number of elements of a finite set is a natural number (non-negative integer), and is called the cardinality of the set.

A set that is not finite is called infinite. For example, the set of all positive integers is infinite: (1,2,3,4, . . .)