Recent content by rumjum

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.