In order to distinguish convergent sequences from divergent sequences, there are two key elements that we need to consider and those are boundary and monotone.

A convergent sequence is a list of elements that are within a boundary of two numbers. It cannot go beyond one number and it also cannot be below the other number. Another property of convergent sequence is monotone which means that it is always increasing or decreasing. Any sequence that is not convergent, diverges.

An example of a convergent sequence in real life is the age of a human being. As we know, a person’s age always increases by one every year. It cannot be higher than 200 (for now) and cannot be lower than zero. Therefore, the age of a human being is a convergent sequence.

Additionally, let us have an example for a divergent sequence. We can talk about the temperature in a day. As we know, the temperature reaches the highest point at noon and reaches the lowest point at night. In other words, it is increasing from night to noon and decreasing from noon to night which means it is not monotone. Therefore, we say the temperature in a day is a divergent sequence.

In conclusion, boundary and monotone are the key terms to distinguish convergent sequences from divergent sequences.