What does the "N" mean in a Cauchy sequence definition? Hi everyone, I have a question regarding Cauchy sequences. I am trying to teach myself real analysis and would appreciate any clarification anyone has regarding my question. I believe I have an intuitive understanding of what a Cauchy sequence means: given any ε>0 the difference between the terms in a sequence approaches zero as the index of the terms (n,m) go to infinity. In other words, as one looks at larger and larger terms, the terms get closer and closer together. (Please pardon that I'm not fluent in Latex). Many of the definitions say something about "...given an ε>0, there exists an N such that..." What does the "N" mean. What is its significance in the definition of a Cauchy sequence? I hope this was explained well enough and I appreciate all help! Thanks!