some a questions in Metric Spaces

cotton candy

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 ...

HallsofIvy

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