Recent content by jetplan

Let's say I have a countable set of real number X = {a1, a2, a3 ... } Define A1 = {a1} A2 = {a1,a2 } ... An = {a1, a2, ... , an} Is it legit to write that \displaystyle\lim_{n\rightarrow +\infty} {A_n} = X ?Is there an ε-δ notation for set operation, equivalent to what we have for sequence...

Hi mfb, Thanks for the cool explanation. So, do you think it makes sense if I say: +++++ In general, "Predicate P(n) is true for all integer n" has nothing to do with the truth value of \\lim_{n\rightarrow +\infty}P(n). We need to apply different tricks specific to the problem...

Hi, Consider the following two logics: Logic 1: Given an infinite countable set of real number A = {a1, a2, a3 ... } If there exist a real number M such that: a1 + a2 + a3 + ... + an < M for all integer n, then the infinite sum a1 + a2 + a3 ... < M Logic 2: Given an infinite collection of...

I agree, and there goes the loophole Thanks !

Sorry if my writing is a little bit rough. The (10-\epsilon, 10 + \epsilon) covers {10} and does not cover mid-point of any other interval. the way we choose \epsilon is arbitrary as long as it doesn't touch the mid-point of other intervals

Not if I construct my Open interval the way I start my post. Compactness require EVERY open cover U possesses a FINITE subcover V which is the subset of the U With U being the way we construct it, i simply can't find a finite V

The point is, each mid-point of the open interval is not covered by any other interval and we have infinitely many of such mid-point. For example, only (0,5) cover {2.5} only (5, 7.5) cover {6.25} only (7.5, 8.75) cover {8.125} only (8.75, 9.375) cover {9.0625} etc etc we have infinitely...

Let's say, for (0, 5) we take (0 - 0.1, 0 + 0.1) to cover the point 0 we take (5 - 0.01, 5 + 0.01) to cover the point 5 we take (7.5 - 0.001, 7.5 + 0.001) to cover the point 7.5 etc etc how can we get a FINITE subcover ?

Yes but that won't be finite. It takes infinitely many of such open interval to cover [0,10] because I cut the length of each interval by half, i.e. length of [0,10] = 10 10 = 5 + 5/2 + 5/4 + 5/8 + 5/16 + ... To show [0,10] is compact we need to find a finite subcover under such construction.

Hi All, So all closed interval [a,b] is compact (see Theorem 2.2.1 in Real Analysis and Probability by RM Dudley) Now, Let's say I have [0,10] as my closed interval. Let My Open Cover be (0, 5) (5, 7.5) (7.5, 8.75) (8.75, 9.375) ... Essentially, The length of each open...

Thanks for the neat proof. one comment: since we know that \sum 1/n! converges, we know e converges because e = 1 + \sum 1/n! where n->infinity

Hi All, How do we go about showing the euler's number e converges ? Recall that e =(1+1/n)^n as n ->infinity Some place prove this by showing the sequence is bounded above by 3 and is monotonic increasing, thus a limit exist. But I forgot how exactly the proof looks like...

Hi All math lovers, I have seen 2 different definition of a neighborhood of a point. Which one is correct ? Given a Topological Space (S,T), a set N \subset S is a neighborhood of a point x \in S iff 1. \exists U \in T, such that x \in U \subseteq N i.e. a neighborhood of a point...

Hi All, I can't see how the following is proved. Given two topological space (X, T), (Y, U) and a function f from X to Y and the following two statements. 1. f is continuous, i.e. for every open set U in U, the inverse image f-1(U) is in T 2. For every convergent filter base F -> x, the...

Thanks everyone for the nice explanation. Anyone care to point out if the following is correct or not ? 1. An open set in a metric space is NEVER a closed set in ANY metric space. Given any set X, d(x,y)= 0 if x= y, 1 if x\ne y is a metric. ALL subsets of X are both open and closed in...