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Cauchy sequence

  • #1
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Homework Statement


Suppose that {[tex]p_n[/tex]} is a Cauchy sequence and that there is a subsquence {[tex]p_{n_i}[/tex]} and a number [tex]p[/tex] such that [tex]p_{n_i} \rightarrow p[/tex]. Show that the full sequence converges, too; that is [tex]p_n \rightarrow p[/tex].


Homework Equations





The Attempt at a Solution


Take [tex]\varepsilon > 0[/tex]. take [tex]N[/tex] s.t. [tex]n_k,n > N[/tex] implies that [tex]d(p_{n_k},p)< \frac{\varepsilon}{2}, d(p_n,p_{n_k}) < \frac{\varepsilon}{2}[/tex]. Hence [tex]d(p_n,p) \leq d(p_{n_k},p)+d(p_{n_k},p_n) \leq \varepsilon[/tex] Thus {[tex]p_n[/tex]} converges to [tex]p[/tex].
 

Answers and Replies

  • #2
radou
Homework Helper
3,115
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Actually, [tex]d(p_n,p) \leq d(p_{n_k},p)+d(p_{n_k},p_n) < \varepsilon[/tex].
 
  • #3
125
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isnt that what i wrote?
 
  • #4
125
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Is the only thing i need to fix the scrictly less than inequality??
 
  • #5
radou
Homework Helper
3,115
6
Is the only thing i need to fix the scrictly less than inequality??
Yes. Without strictly less, you're not consistent with the definition of convergence.
 

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