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Some a questions in Metric Spaces

  1. Apr 17, 2008 #1
    Hii
    my Dears....

    I'm new student in the Math & I'm so bad in the English Language..
    But, I want to learn this language ...


    to excuse me ...

    I have some a questions about Metric Spaces ..

    Q1:If (X,d) is a metric spaces . Prove the fallwing:

    1* ld(x,y)-d(z,y)l [tex]\leq[/tex] d(x,y)+d(y,w).
    2* ld(x,z)-d(y,z)l [tex]\leq[/tex] d(x,y). ..??



    Q2:Prove that:
    Xn ـــــــــ> X iff [tex]\forall[/tex] V (neighborhood of X) [tex]\exists[/tex] n0 is number s.t Xn[tex]\in[/tex]V [tex]\forall[/tex]n>n0....??


    Q3: If (X1,d1) & (X2,d2) is a metrics spaces, Prove that X=X1xX2 () is a metric spaces whith a metric defind by: d(x,y)=d1(x1,y1)+d2(x2,y2) s.t x1,y1[tex]\in[/tex]X1..??


    Q4:Prove that every Cauchy sequence in a metric space (X, d) is bounded...??


    I need to help by speed..
    Thanx ...
     
    Last edited: Apr 17, 2008
  2. jcsd
  3. Apr 17, 2008 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Using what basic definitions, postulates, etc.?



    What is your definition of "Xn ـــــــــ>X"?


    Show that the conditions for a metric space are satisfied- in other words what is the definition of "metric space".


    If {xn} is a Cauchy sequence, then there exist N such that if both m, n> N, d(xn, xm)< 1. Let M= largest of d(xn,xm) for n and m [itex]\le[/itex] N+1. Can you prove that d(xn,xm)[itex]\le[/itex] M+ 1 for all m and n.

    I need to help by speed..
    Thanx ...[/QUOTE]

    In mathematics, definitions are working definitions. You use the specific words of definitions in proofs.
     
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