A Cauchy sequence is a sequence of numbers in which the terms become arbitrarily close to each other as the sequence progresses. In other words, for any small positive number, there exists a point in the sequence after which all the terms are within that small distance from each other.
To show that x_n is Cauchy, we need to prove that for any small positive number, there exists a point in the sequence after which all the terms are within that small distance from each other. This can be done by using the definition of a Cauchy sequence and applying it to the given sequence x_n.
Showing that x_n is Cauchy is important because it guarantees that the sequence is convergent. This means that the sequence has a limit and the terms of the sequence become arbitrarily close to this limit as the sequence progresses. This is a fundamental concept in analysis and has many applications in mathematics and other fields.
No, a Cauchy sequence cannot be divergent. This is because a Cauchy sequence is defined as a sequence in which the terms become arbitrarily close to each other as the sequence progresses. If the sequence were to diverge, the terms would become increasingly farther apart and would not satisfy the definition of a Cauchy sequence.
The Cauchy criterion is a fundamental concept in real analysis and is used to prove the convergence of sequences and series. It is also used to define important concepts such as completeness and continuity. Additionally, the Cauchy criterion is used in the construction of the real numbers from the rational numbers.