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!
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...
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!
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...
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 ...
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.
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...
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) -...
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.
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.
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)...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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.
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'...
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...