Exploring Convergence in Baby Rudin: Understanding the Sequence {1/n}

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In summary, on page 48 of baby Rudin, it is stated that the sequence {1/n} converges in R1 to 0, but fails to converge in the set of all positive real numbers [with d(x,y) = |x-y|]. This is because the positive reals do not include zero, which is the limit of the sequence.
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henryN7
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on page 48 of baby Rudin, it says " the sequence {1/n} converges in R1 to 0, but fails to converge in the set of all positive real numbers [with d(x,y) = |x-y|]."

ok, I know it has something to do with 1/n going to infinity near zero, but it does that whether the metric space is R1 or just the positive reals.

So why is that quote true?
 
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  • #2
henryN7 said:
on page 48 of baby Rudin, it says " the sequence {1/n} converges in R1 to 0, but fails to converge in the set of all positive real numbers [with d(x,y) = |x-y|]."

ok, I know it has something to do with 1/n going to infinity near zero, but it does that whether the metric space is R1 or just the positive reals.

So why is that quote true?

Because the positive reals do not include zero, the limit of the sequence.
 
  • #3
awkward said:
Because the positive reals do not include zero, the limit of the sequence.

Ah, thank you.
 

FAQ: Exploring Convergence in Baby Rudin: Understanding the Sequence {1/n}

What is convergence and how does it relate to scientific research?

Convergence is the process of combining multiple disciplines and methods to address complex scientific problems. It allows researchers to approach problems from a variety of angles and often leads to innovative solutions.

Why is convergence important in scientific fields?

Convergence fosters collaboration and cross-disciplinary communication, leading to breakthroughs that may not have been possible with traditional research methods. It also allows for a more comprehensive understanding of complex systems and phenomena.

Can you provide an example of convergence in action?

An example of convergence in action is in the field of nanotechnology, which combines principles from physics, chemistry, and biology to create new materials and devices with unique properties and applications.

What challenges are associated with convergence in scientific research?

One of the main challenges is breaking down traditional barriers between disciplines and encouraging collaboration and communication between researchers from different backgrounds. There can also be difficulties in securing funding for cross-disciplinary projects.

How can scientists incorporate convergence into their research?

Scientists can incorporate convergence into their research by actively seeking out collaborations with researchers from different disciplines, attending conferences and workshops that focus on convergence, and being open to new and innovative approaches to problem-solving.

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