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 $$\leq$$ d(x,y)+d(y,w).
2* ld(x,z)-d(y,z)l $$\leq$$ d(x,y). ..??



Q2:Prove that:
Xn ـــــــــ> X iff $$\forall$$ V (neighborhood of X) $$\exists$$ n0 is number s.t Xn$$\in$$V $$\forall$$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$$\in$$X1..??


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


I need to help by speed..
Thanx ...

 

[HallsofIvy]

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 $$\leq$$ d(x,y)+d(y,w).
2* ld(x,z)-d(y,z)l $$\leq$$ d(x,y). ..??

Using what basic definitions, postulates, etc.?

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

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

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$$\in$$X1..??

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

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

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 $\le$ N+1. Can you prove that d(x_n,x_m)$\le$ 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.