Introduction to the epsilon-delta definition of limits

In accordance with many standard textbooks of calculus, we explain the epsilon-N definition of the limits of sequences before the epsilon-delta definition of the limits of functions. This is the best way to understand the essence of the epsilon-delta definition of limits.

Chapter 1 Epsilon-N Definition

First, we explain the underlying basic idea of the epsilon-N definition of the limits of sequences through the parable of an airplane approaching an airport. Next, we introduce the epsilon-N definition of the limits of sequences, which lays down the basis for rigorous studies of the limits of sequences. Finally, we show simple examples in which the epsilon-N definition is used in arguments concerning convergent sequences.

Chapter 2 Some Basic Techniques

We explain the triangle inequality, logic symbols, and the negation of statements, which are basic techniques used in arguments based on the epsilon-N argument. In order to apply the epsilon-N definition to concrete problems, it is necessary to become familiar with these techniques. We note that they are used in arguments based on the epsilon-delta definition of the limits of functions.

Chapter 3 Epsilon-Delta Definition

We can analogously understand the epsilon-delta definition of the limits of functions if we understand the epsilon-N definition of the limits of sequences. We introduce the epsilon-delta definition of the limits of functions, which lays down the basis for rigorous studies of the limits of functions. Next, we use the epsilon-delta definition to propose a rigorous definition of the continuity of functions. In addition, we explain the infimum and supremum of a set of real numbers in the same arguments that use the epsilon-delta definition to investigate the limits of functions.