How do you prove that a sequence is a Cauchy sequence?

How do you prove that a sequence is a Cauchy sequence?

A sequence {an}is called a Cauchy sequence if for any given ϵ > 0, there exists N ∈ N such that n, m ≥ N =⇒ |an − am| < ϵ. |an − L| < ϵ 2 ∀ n ≥ N. Thus if n, m ≥ N, we have |an − am|≤|an − L| + |am − L| < ϵ 2 + ϵ 2 = ϵ.

How do you prove a Cauchy sequence in the metric space?

Theorem 1.11 – Convergent implies Cauchy In a metric space, every convergent sequence is a Cauchy sequence. d(xn,x) < ε/2 for all n ≥ N. Using this fact and the triangle inequality, we conclude that d(xm,xn) ≤ d(xm,x) + d(x, xn) < ε for all m, n ≥ N. This shows that the sequence is Cauchy.

How do you show two Cauchy sequences are equivalent?

Two Cauchy sequences are called equivalent if and only if the difference between them tends to zero. This defines an equivalence relation that is compatible with the operations defined above, and the set R of all equivalence classes can be shown to satisfy all axioms of the real numbers.

Is (- 1 N Cauchy sequence?

Think of it this way : The sequence (−1)n is really made up of two sequences {1,1,1,…} and {−1,−1,−1,…} which are both going in different directions. A Cauchy sequence is, for all intents and purposes, a sequence which “should” converge (It may not, but for sequences of real numbers, it will).

Is every Cauchy sequence convergent in metric space?

Sets, Functions and Metric Spaces Every convergent sequence {xn} given in a metric space is a Cauchy sequence. If is a compact metric space and if {xn} is a Cauchy sequence in then {xn} converges to some point in . In n a sequence converges if and only if it is a Cauchy sequence.

What is equivalence class of Cauchy sequence?

A representative of an equivalence class of Cauchy sequence is any Cauchy sequence belonging to that class. Real Numbers: A real number is defined to be an equivalence class of 1 Page 2 Cauchy sequences. Each real number is a different equivalence class. The set of all real numbers is denoted R.

How do you define reals?

The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as (1.41421356…, the square root of 2, an irrational algebraic number). Included within the irrationals are the real transcendental numbers, such as π (3.14159265…).