A Cauchy sequence is a sequence of numbers where the terms become arbitrarily close to each other as the sequence progresses. In simpler terms, for any small distance you choose, there comes a point in the sequence where all subsequent terms are within that distance from each other. This property indicates that the sequence is converging towards a limit, even if that limit is not explicitly known. Cauchy sequences are important in real analysis and metric spaces because they help establish the completeness of a space. A space is considered complete if every Cauchy sequence within it converges to a limit that is also within the same space. This concept is fundamental in understanding the behavior of sequences and functions in mathematics.