Definition of cauchy sequence8/10/2023 ![]() ![]() If it was, perhaps a reminder somewhere in that section would have been good. I would check the Preface or previous chapters to see if it was mentioned that this restriction was the case. But you are correct that the statement is not generally true for analysis over all spaces. In the case of the real line, every Cauchy sequence converges that is, being a Cauchy sequence is sufficient to guarantee the existence of a limit. ![]() Once you get far enough in a Cauchy sequence, you might suspect that its terms will start piling up around a certain point because they get closer and closer to each other. Every convergent sequence is a Cauchy sequence. I like the way an old professor of mine described it:Īll Cauchy sequences want to converge to something, but if it can't it's not the fault of the sequence, it's a fault of the space.įor example, the sequence $\$ - which is complete. Theorem 5 (Cauchy Sequences) Two important theorems: 1. ![]() But the converse is only true for certain spaces. The answer is that it depends on the space!Ĭertainly, if a sequence converges it is Cauchy (regardless). ![]()
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