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...