# Search results

1. ### Set of continuous bounded functions.

Yes, the Heine-Borel theorem and compactness is clear to me. I do have a question though about the limit of f_n "not being" a limit in any metric space. This part is not clear to me. Thanks again though!
2. ### Set of continuous bounded functions.

This is exactly what I had on my mind. Thanks for clarifying the same. It is very helpful. I understood the part of boundedness and the fact that the set of functions {x^n} is also not compact as there is a sequence {fn} which has limit point {f} that is not in C(X). The only confusion was about...
3. ### Set of continuous bounded functions.

I know what that closed set have all its limit points. You don't need to explain that part to me. By the way, I think you should read rudinreader's comments. They are right on dot!
4. ### Set of continuous bounded functions.

It forms a closed sphere with radius =1. Also, the limit points of the sequence of functions is either 0 or 1 and both are contained in the range of fn(x). Yes, I stand corrected. I realized that the sequence of functions are all continuous. It is just the f(x) to which the sequence tends to...
5. ### Set of continuous bounded functions.

Homework Statement This is not a homework question. I am solving this from the lecture notes that one of my friend's has got from last year. If C(X) denotes a set of continuous bounded functions with domain X, then if X= [0,1] and fn(x) = x^n. Does the sequence of functions {fn} closed ...
6. ### Pointwise vs. Uniform Convergence.

I re-read your statement on pointwise convergence of the previous post and referred to the text book. I understood what you were saying about pointwise convergence. Thanks for further clarifying the same and for your example. Appreciate it.
7. ### Pointwise vs. Uniform Convergence.

Well, I understood the part of finding the sup using derivatives. And yes you are right that the domain is set of Real number (R). But, now I am a bit more confused about the difference between pointwise and uniform convergence. I am under the impression (and correct me if am wrong) that the...
8. ### Pointwise vs. Uniform Convergence.

Homework Statement I need to understand as to why the following series fn(x) = x/(1+n*x^2) is point wise convergent (as mentioned in the book of Ross) and not [obviously] uniformly convergent. Homework Equations The relevant equation used is that lim (n-> infinity) sup|(fn(x) -...
9. ### Every sequence of bounded functions that is uniformly converent is uniformly bounded

That is an important point that you brought up. Thanks, for that. I solved the problem by showing that |f(x)| < M(N+1)+1 for e=1 and |fn| < Mn. And, since for n >=N, the function is uniformly bounded, we have |f(x)| < 1 + M(N+1). Henc,e |f(x)| is bounded. Thanks, again.
10. ### Every sequence of bounded functions that is uniformly converent is uniformly bounded

That is an important point that you brought up. Thanks, for that. I solved the problem by showing that |f(x)| < M(N+1)+1 for e=1 and |fn| < Mn. And, since for n >=N, the function is uniformly bounded, we have |f(x)| < 1 + M(N+1). Henc,e |f(x)| is bounded. Thanks, again.
11. ### Every sequence of bounded functions that is uniformly converent is uniformly bounded

Homework Statement Prove that every sequence of bounded functions that is uniformly convergent is uniformly bounded. Homework Equations Let {fn} be the sequence of functions and it converges to f. Then for all n >= N, and all x, we have |fn -f| <= e (for all e >0). ---------- (1)...
12. ### Geometric interpretation of Generalized MVT

Homework Statement I am trying to see the geometric interpretation of the generalized MVT. It is not a homework problem, but would like to know how to interpret the equation Homework Equations [f(b)- f(a)]* g'(x) = [g(b)- g(a)]* f'(x) The Attempt at a Solution On...
13. ### Simple question on continuity

Well, how is this solution then. I am badly confused. So please go through this one. If X is bounded non empty subset in R (usual) and f:X->R is uniformly continuous function. Prove that f is bounded. Since X is bounded in R, it has a supremum and infimum. Also, we can have a...
14. ### Simple question on continuity

Homework Statement If X is bounded non empty subset in R (usual) and f:X->R is uniformly continuous function. Prove that f is bounded. Homework Equations The Attempt at a Solution Since X is bounded in R, it is a subset of cell. And all cells in R are compact.All bounded sub...
15. ### A simple problem in Real Analysis

Thanks for the detailed response. Really appreciate it.
16. ### Simple Question on continuity

Homework Statement 1) If f is a continuous mapping from a matric space X to metric space Y. A E is a subset of X. The prove that f(closure(E)) subset of closure of f(E). 2) Give an example where f(closure (E)) is a proper subset of closure of f(E). Homework Equations The...
17. ### Simple question in topology (finite vs. infinite)

Homework Statement I always get confused between countably many vs. uncountable. I suppose if one can index the points of a set , then it is countable. 1)So, anything that is finitie is countable. Anything that is infinite is also countable? Then what is uncountable, something that...
18. ### Very badly stuck (Prove that X is compact)

Thank you very much!
19. ### Very badly stuck (Prove that X is compact)

I think I do see what you are saying. If I understand this correctly, "p" is in one of the Gn's as you said and Gn is open. Hence, all points of Gn are internal points. In other words, one can find a neighborhood Nr(p) such that it is a subset of a Gn. But E contains points of Fn that are not in...
20. ### Very badly stuck (Prove that X is compact)

Homework Statement If X is a metric space such that every infinite subset has a limit point, then prove that X is compact. Homework Equations Hint from Rudin: X is separable and has a countable base. So, it has countable subcover {Gn} , n=1,2,3..... Now, assume that no finite sub...
21. ### Please comment. (Topology question on separability)

Homework Statement If X be a metric space in which every infinite subset has a limit point, then X is separable. This is a question from Rudin but I am having some difficulty just understanding how to use the hint. Homework Equations The hint as in the book is . Fix delta >0, and...
22. ### A simple problem in Real Analysis

No problem. Have a good day!
23. ### A simple problem in Real Analysis

Thank you very much to start with. I think I now understand the problem better. Just before you posted your approach, I had mine as follows. I think a part of my approach is similar to your. But yours is much more cleaner. I will appreciate if you can read the following and understand my...
24. ### A simple problem in Real Analysis

Homework Statement I am having somewhat a difficult time just understanding a simple concept. I am trying to prove that every open subset G of a separable metric space X is the union of a sub collection {Vi} such that for all x belongs to G, x belongs to some Vi (subset of G). I am...
25. ### A metric space having a countable dense subset has a countable base.

I suppose I wanted to say that the collection of Vi is countable and not finitely many. Anyways...
26. ### A metric space having a countable dense subset has a countable base.

1. Homework Statement Every separable metric space has countable base, where base is collection of sets {Vi} such that for any x that belongs to an open set G (as subset of X), there is a Vi such that x belongs to Vi. 2. Homework Equations Hint from the book of Rudin: Center the point...
27. ### Every separable Metric space has countable base.

Homework Statement Every separable metric space has countable base, where base is collection of sets {Vi} such that for any x that belongs to an open set G (as subset of X), there is a Vi such that x belongs to Vi. Homework Equations Hint from the book of Rudin: Center the point in a...
28. ### Prove that the closure is the following set.

Does not make any sense to me Edit: Yup, it makes sense now. For some reason, the class on sequence is chapter 3 in Rudin but the homework problem given to me is after Chapter 2. so, it was not not easy understanding the subsequence part. However, I solved it in a different way.
29. ### Prove that the closure is the following set.

Homework Statement Suppose (X,d) is a metric space. For a point in X and a non empty set S (as a subset of X), define d(p,S) = inf({(d(p,x):x belongs to S}). Prove that the closure of S is equal to the set {p belongs to S : d(p,S) =0} Homework Equations Closure of S = S U S' , where S'...
30. ### Open n-cell is open?

I understand that to prove a point x as internal point, I do not need to prove all points are internal points. I can just take x and find a neigborhood of x that is contained in the cell. And if all points of cell are internal points, then the set/cell is open. If that was not apparent from my...