Sequences and Nets: Does (1/n) Converge to [0,1]?

  • Context: Graduate 
  • Thread starter Thread starter ForMyThunder
  • Start date Start date
  • Tags Tags
    Sequences
Click For Summary

Discussion Overview

The discussion revolves around the convergence of the sequence (1/n) within a specific topology on the real numbers, particularly focusing on the neighborhood properties of the interval [0,1]. Participants explore whether (1/n) converges to every point in [0,1] under the proposed topological structure.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant suggests that if the real numbers are given a topology \left\{R,\phi, [0,1]\right\}, then (1/n) converges to every point of [0,1] since it is a neighborhood of every point.
  • Another participant questions the definitions of R and \phi, asking whether the proposed topology satisfies the definition of a topology.
  • A different participant asserts that if \mathbb{R} has the topology \{\emptyset,[0,1],\mathbb{R}\}, then (1/n) converges to every point of \mathbb{R}.
  • A later reply acknowledges the previous point about convergence in the specified topology, indicating some level of agreement.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the proposed topology and its implications for convergence. There is no consensus on the correctness of the topology or the convergence claims.

Contextual Notes

Limitations include potential ambiguities in the definitions of the topology and the sequence's convergence properties, as well as the need for clarification on the topology's adherence to formal definitions.

ForMyThunder
Messages
149
Reaction score
0
Say the real numbers were given a topology \left\{R,\phi, [0,1]\right\}. Does the sequence (1/n) converge to every point of [0,1] since it is a neighborhood of every point?
 
Last edited:
Physics news on Phys.org
What is R and \phi? Does your topology satisfy the definition of a topology?
 
If \mathbb{R} has the topology \{\emptyset,[0,1],\mathbb{R}\}, then the sequence (1/n) converges to every point of \mathbb{R}!
 
micromass said:
If \mathbb{R} has the topology \{\emptyset,[0,1],\mathbb{R}\}, then the sequence (1/n) converges to every point of \mathbb{R}!

Yeah, I guess you're right. Thanks.
 

Similar threads

Graduate Constructing a sequence in a manifold
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad (0,1) is uncountable using binary expansions?
  • · Replies 15 ·
Replies
15
Views
4K
Undergrad Question about uniform convergence in a proof
  • · Replies 1 ·
Replies
1
Views
3K
Prove that the sequence does not have a convergent subsequence
  • · Replies 4 ·
Replies
4
Views
2K
Unit normed linear functional on a space of sequences
  • · Replies 13 ·
Replies
13
Views
2K
Bounded non-decreasing sequence is convergent
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Convergence not defined by any metric
  • · Replies 17 ·
Replies
17
Views
2K
MHB Uniform convergence - Length of graph
  • · Replies 21 ·
Replies
21
Views
3K
Graduate Why does the characteristic function not converge in the L1 space?
  • · Replies 11 ·
Replies
11
Views
5K
MHB How can we prove the convergence of recursive defined sequences?
  • · Replies 2 ·
Replies
2
Views
2K