Real Analysis refers to the study of real valued functions. It particularly focuses on the properties of these functions, including but not limited to, the convergence and limits of sequences of real numbers, as well as the calculus of real numbers. In dealing with Real Analysis, it is important to understand the different terms. If a limit for a particular sequence exists, then the sequence is called convergent; however if the limit for a particular sequence does not exist, then the sequence is called divergent.
For example, if the sequence was 1, ½, 1/3 , ¼, 1/5, 1/6 and so on, this can be written in the following form:
j is the denominator of the function
inf is infinity
This sequence eventually converges to zero, because as j approaches infinity, the denominator gets extremely large, leaving the output of the function to be extremely small. However, the important aspect to consider is that the series never actually reaches zero, but it can get as infinitesimally close to it. This is why the study of limits in this particular context is considered under Real Analysis.