set of all finite subsets of a continuum has cardinality continuum
Hi. I'm learning about equivalent sets in a real analysis class and am struggling a bit with the abstractness of it. Would someone be able to explain in very basic language how to:
Prove that the set of all finite subsets of a continuum has the cardinality of continuum.
I guess I'm not totally sure what a continuum is. Is it just any set with a bijection between it and the set of real numbers?
From what I know a set has cardinality of continuum if it is equivalent to the set of all sequences whose elements are the digits 0 and 1 (which is equivalent to the set of real numbers).
So I need a way to write each finite subset as a sequence of 1s and 0s to create a bijection between it and the set of all sequences whose elements are the digits 0 and 1?
I'm definitely confused about where to start, and would really appreciate some help!