manooba said:Let (Xn) be a sequence satisfying
|Xn+1-Xn| ≤ λ^n r
Where r>0 and λ lies between (0,1). Prove that (Xn) is a Cauchy sequence and so is convergent.
A Cauchy sequence is a sequence of numbers in which the terms get closer and closer together as the sequence progresses. In other words, for any positive number ε, there exists a point in the sequence after which all the terms are within ε distance of each other.
To prove that a sequence is Cauchy, you must show that for any positive number ε, there exists a point in the sequence after which all the terms are within ε distance of each other. This can be done by using the definition of a Cauchy sequence and manipulating the terms of the sequence until the desired result is achieved.
Proving a sequence to be Cauchy is important because it shows that the sequence is convergent. This means that the terms of the sequence get closer and closer to a certain limit as the sequence progresses, making it easier to analyze and work with mathematically.
A Cauchy sequence is a necessary but not sufficient condition for convergence. This means that if a sequence is Cauchy, it must be convergent, but a sequence can be convergent without being Cauchy. However, in many cases, proving a sequence to be Cauchy is the first step in proving its convergence.
One example of a Cauchy sequence is the sequence {1, 1/2, 1/3, 1/4, ...}. Its limit is 0, as the terms get closer and closer to 0 as the sequence progresses. This can be proven using the definition of a Cauchy sequence by choosing any positive number ε and showing that there exists a point in the sequence after which all the terms are within ε distance of each other.