1. shevron

let $\left( X;d \right)$ be a matric space and $C_{b}$$\left( X,R \right)$ denote the set of all continuous bounded real valued functions defined on X, equipped with the uniform metric. $d\left( f,g \right)=sup{ \left| f \left( x \right)-g \left( x \right) \right|: xinX }$ Show that $C_{b}$$\left( X,R \right)$ is a complete matric space

2. shevron

3. timo86m

sorry idk this one :(

4. shevron

@timo86m

5. shevron

6. shevron

@Chlorophyll

7. shevron

@charliem07

8. charliem07

sorry i dont know

9. shevron

ok cool

10. shevron

11. shevron

12. shevron

13. UnkleRhaukus

@JamesJ, @experimentX, @eliassaab, @nbouscal, @beketso

14. shevron

15. TuringTest

@KingGeorge

16. TuringTest

btw for advanced questions you may have better luck here http://math.stackexchange.com/

17. shevron

k thanx

18. satellite73

your job is showing it is "complete" is to show that if $$f_n\to f$$ then $$f\in C_b$$

19. satellite73

that is, if a sequence of continuous functions converges to some function using the sup metric, then the limit function is continuous also

20. satellite73

this should work because the metric is the supremum over all $$x$$

21. satellite73

the general idea is that under the sup metric, the convergence is uniform, and the uniform limit of a sequence of continuous functions is uniform gotta run, but if you google what i wrote i bet you will find a worked out solution

22. shevron

do i have to let the sequence to be a cauchy sequence first?

23. satellite73

actually what i meant is the uniform limit of a sequence of continuous functions is CONTINUOUS

24. satellite73

yes

25. shevron

my problem we are given f and g and they are different how am i going to proof them simultaneously

26. shevron

@Mertsj

27. shevron

@ash2326

28. shevron

@walters

29. shevron

30. phi

I assume you mean "metric space" ? But I tend more to applied math problems. i.e. not this kind of question.

31. shevron

ok cool

32. shevron

can u fynd me someone who can do it

33. shevron

34. shevron

35. mathslover

Sorry, am not good at this topic.

36. shevron

ok can you search for me where i can find something related to ths?

37. mathslover

Yes! I am best at that field :)

38. shevron

39. mathslover

40. shevron

eish they blocked youtube here at school

41. mathslover
42. mathslover
43. shevron

ok thanx

44. mathslover

Have a look at the links and let me know whether they helped or not.

45. shevron

ok i will

46. shevron

@mathslover they are not helping

47. mathslover
48. shevron