A Cauchy sequence is a sequence of numbers where the terms get closer and closer together as the sequence progresses. In other words, for any positive number, there exists a point in the sequence where all the terms after it are within that distance of each other.
To prove a sequence is Cauchy, you need to show that for any positive number, there exists a point in the sequence where all the terms after it are within that distance of each other. This can be done using the definition of a Cauchy sequence and using epsilon-delta proofs.
An epsilon-delta proof is a mathematical proof used to show that a function or sequence converges to a certain value. It involves choosing a small value of epsilon (ε) and showing that for any delta (δ), the function or sequence will be within ε units of the desired value at all points within δ units of the original point.
No, a sequence cannot be both Cauchy and divergent. A Cauchy sequence must converge to a limit, while a divergent sequence does not have a limit. Therefore, a sequence cannot exhibit both behaviors simultaneously.
No, not all Cauchy sequences are convergent. A Cauchy sequence may converge to a limit, but it is not guaranteed. In order for a sequence to be convergent, it must also be bounded, which is not a requirement for a Cauchy sequence. Therefore, some Cauchy sequences may not have a limit and are not convergent.