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Convergent Sequences

  • Thread starter Kreizhn
  • Start date
  • #1
743
1

Homework Statement


Let (X,d) be a metric space with two sequences [itex] (x_n), (y_n) [/itex] which converge to values of a,b respectively. Show that

[tex] \lim_{n \to \infty} d(x_n,y_n) = d(a,b) [/tex]


Homework Equations


[tex] (x_n) \rightarrow a \Leftrightarrow \forall \epsilon >0 \quad \exists n_0 \in \mathbb{N} \text{ such that } \forall n>n_0 \quad d(x_n,a)< \epsilon [/tex]

[tex] d(x,z) \leq d(x,y) + d(y,z) \quad \forall x,y,z \in X [/tex]


The Attempt at a Solution



This seems like it should be a fairly easy question, but I don't have much analysis in my background. I attempted to proceed as follows:

Since [itex] d:X\times X \rightarrow \mathbb{R} [/itex], it is sufficient to show that [itex] \forall \epsilon >0 \quad \exists n\in \mathbb{N} \text{ such that } |d(x_n,y_n) - d(a,b)|< \epsilon [/itex]. So let [itex] \epsilon >0 [/itex] and [itex] n' = max\{ n_0, n_1 \} [/itex] where [itex] n_0, n_1 [/itex] are natural numbers which satisfy the individual convergence properties for [itex] (x_n),(y_n)[/itex]. Let [itex] n>n' [/itex] giving

[tex] |d(x_n,y_n) - d(a,b) | &=& |d(x_n,y_n) + d(x_n,b) - d(x_n,b) - d(a,b)|

\leq |d(y_n,b) - d(x_n,a)|

< |\epsilon - \epsilon|

\leq \epsilon[/tex]

But I'm really not sure about the [itex] |\epsilon - \epsilon| \leq \epsilon [/itex] line.

Any thoughts would be appreciated.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
Well, if 0<=e1<=e and 0<=e2<=e then |e1-e2|<=e. That's what you really mean by |e-e|<=e, right?
 
  • #3
743
1
Yes, that was my thought process.
 

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