Help with Spivak's treatment of epsilon-N sequence definition

Overall, the use of epsilon-N proofs in sequence convergence is a rigorous method that can be used to prove convergence without directly appealing to the definition. However, there may be alternative methods that can also be used to prove convergence. In summary, the use of epsilon-N proofs in sequence convergence is a rigorous method that can be used to prove convergence without directly appealing to the definition, but there may be alternative methods available as well.
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mitcho
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I have just started my first real analysis course and we are using Spivak's Calculus. We have just started rigorous epsilon-N proofs of sequence convergence. I was trying to do some exercises from the textbook (chapter 22) but there doesn't seem to be any mention of epsilon-N in the solutions (apart from the first one). Are they still rigorous proofs or what? Can you rigorously prove that a sequence converges without appealing to the definition?

It would also be good if someone who had the text could take a quick look at the solutions to the exercises at the end of chapter 22 (third edition) and see what they think about them.

Thanks
 
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  • #2
you don't always have to appeal directly to the definitions. For example, if you have a non-negative sequence of numbers { a_n }, and if you furthermore know that the non negative sequence of numbers { b _ n } converges to zero* with a_ n < b _ n , then it is clear that {a_n } converges ( of course, the proof of this theorem would have required some delta-epsilon proof ).

Or, if you have an increasing sequence that is bounded, then it converges.
 
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FAQ: Help with Spivak's treatment of epsilon-N sequence definition

What is Spivak's treatment of epsilon-N sequence definition?

Spivak's treatment of epsilon-N sequence definition is a way of formally defining the limit of a sequence in calculus. It involves using the concept of epsilon, which represents a small positive number, and N, which represents a large positive integer, to prove that a given sequence approaches a specific limit.

Why is Spivak's treatment of epsilon-N sequence definition important?

Spivak's treatment of epsilon-N sequence definition is important because it provides a rigorous and precise way to define limits in calculus. Without a formal definition, it can be difficult to prove the existence or non-existence of a limit, which is essential in many areas of mathematics and science.

What is the process for using Spivak's treatment of epsilon-N sequence definition?

The process for using Spivak's treatment of epsilon-N sequence definition involves first stating the limit definition using epsilon and N, and then using algebraic and logical manipulations to prove that the given sequence satisfies the definition. This usually includes setting up an inequality involving epsilon and N, and then finding a value for N that guarantees the inequality is true for all values of epsilon.

Are there any limitations to Spivak's treatment of epsilon-N sequence definition?

While Spivak's treatment of epsilon-N sequence definition is a powerful tool for defining limits, it does have some limitations. It may not work for all types of sequences, and in some cases, alternate methods may need to be used to prove the existence or non-existence of a limit. Additionally, the process can be quite complex and time-consuming for more complicated sequences.

How can I improve my understanding of Spivak's treatment of epsilon-N sequence definition?

To improve your understanding of Spivak's treatment of epsilon-N sequence definition, it is important to practice using it on a variety of sequences. You can also review the underlying concepts of calculus, such as limits, continuity, and convergence, to strengthen your understanding. Additionally, seeking help from a tutor or joining a study group can also be helpful in clarifying any confusion and improving your skills in using this technique.

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