The Cauchy Criterion is a fundamental concept in real analysis that helps determine the convergence of sequences. According to this criterion, a sequence is convergent if, for every small positive number ε, there exists a point in the sequence beyond which the terms are all within ε of each other. This means that as you progress through the sequence, the terms get arbitrarily close together. This criterion is particularly useful because it allows us to assess convergence without needing to know the limit of the sequence. It applies not only to real numbers but also to sequences in metric spaces, making it a versatile tool in mathematical analysis.