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Math Help - Cauchy Sequence

  1. #1
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    Cauchy Sequence

    Dear Colleagues,

    Could you please help me in solving the following problem:

    If (x_{n}) is a Cauchy sequence in the metric space (X,d) and (y_{n}) in X such that d((x_{n}),(y_{n}) \longrightarrow 0 as  n\longrightarrow \inftythen show that  (y_{n}) is Cauchy in X  .
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  2. #2
    Behold, the power of SARDINES!
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    Have you tried drawing a 2d diagram? (they can be great models for metric spaces) Remember since the x_n's are Cauchy you can find an epsilon ball that the tail never leaves.

    Now since the y_n's get close to the x_n's can find another N such that this distance also falls in another epsilon ball.

    As I said draw a picture and what fraction of epsilon you need.

    Hint you will need 3 balls.
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  3. #3
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    Note that for all positive number  \epsilon , you can find N \in \mathbb {N} such that  d(x_n,x_m) < \frac { \epsilon }{3} , likewise for  d(x_n,y_m)

    Now, consider  d(y_n,y_m) \leq d(y_n,x_n) + d(x_n,y_m) \leq d(y_n,x_n) + d(x_n,x_m)+d(x_m,y_m)
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