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Analogous definitions can be given for sequences of natural numbers, integers, etc. Often sequences such as these are called real sequences, sequences of real numbers or sequences in R to make it clear that the elements of the sequence are real numbers. While we are all familiar with sequences, it is useful to have a formal definition.ĭefinition A sequence of real numbers is any function a : N→ R. Get access to the latest Lecture 32: Cauchy Sequence Theorems (in Hindi) prepared with IIT-JAM course curated by Upendra Yadav on Unacademy to prepare. Sequences occur frequently in analysis, and they appear in many contexts. 5.1 Theorem (Limit Superior and Inferior).4.2 Theorem (Nested intervals property).4.1 Theorem (Convergence of Monotone sequences).But many Cauchy sequences do not have multiplicative inverses.

So Cauchy sequences form a commutative ring. The constant sequences 0 (0 0 :::) and 1 (1 1 :::) are additive and multiplicative identities, and every Cauchy sequence (x n) has an additive inverse ( x n). 3.2 Theorem (Squeeze/Sandwich Limit Theorem) Thus we can add and multiply Cauchy sequences.Then there exists N2N such that ja n Lj< 2 8n N: Thus if n m N, we have ja n a mj ja n Lj+ja m Lj< 2 + 2 : Thus fa ngis Cauchy. This is read xn approaches a as n approaches. If ( xn) converges to a then we say a is the limit of ( xn) and write.

if for all >0, there exists N in N such that xn - a < for all n N. The sequence ( xn) is said to converge to a real number a. Let fa ngbe a sequence such that fa ngconverges to L(say). Definition Let ( xn) be a sequence of real numbers. ()) Suppose that limn1 xn x 2 R: We want to show that fxng is Cauchy sequence. Cauchy Convergence Criterion A sequence of real numbers is convergent if and only if it is a Cauchy sequence.