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Jun
12
comment Limits of Subsequences
With the same idea it actually follows that for any subsequence of $\{t_{n}\}$ we find such a subsequence of $\{s_{n}\}$ that decreases more rapidly. And I don't think we needed the fact that $s_{n}\leq t_{n}$ at all.
Jun
12
comment subsets probability question
What have you tried so far? And is this a homework?
Jun
11
comment True if sigma-compact
@SamL.: Thank you.
Jun
11
answered True if sigma-compact
Jun
11
comment Form of $\sigma(X_n)$ and $\sigma(X_n)$-measurability
@Justin: Topology is just the family of open sets. If it confuses you, you can ignore that part and simply think of $\mathscr{B}(X)$ being the Borel $\sigma$-algebra of $X$ the way you have it defined.
Jun
11
comment True if sigma-compact
Found it now. And it's true that this is very questionable since no proof is given. If $X$ is $\sigma$-compact then the Lemma is true, as $E$ is a countable union of Borel sets. Let me see if I'm able to do something with the locally compact Hausdorff case.
Jun
11
comment True if sigma-compact
I couldn't find this Lemma from Royden's book. What page is it?
Jun
11
comment is there any one one and onto function from $A$ to $B$
That is correct.
Jun
11
comment is there any one one and onto function from $A$ to $B$
You could start by writing down the sets $A$ and $B$ in simpler forms. They are both open intervals.
Jun
11
comment True if sigma-compact
What do we know about the topology of $X$?
Jun
11
revised Form of $\sigma(X_n)$ and $\sigma(X_n)$-measurability
added 1 characters in body
Jun
11
answered Form of $\sigma(X_n)$ and $\sigma(X_n)$-measurability
Jun
10
comment Evaluating $\int \frac{dx}{x^2 - 2x} dx$
@MarkBennet: The latter part to the answer was included after all the previous comments took place. And the first part, it is purely unnecessary steps to obtain $\frac{1}{x(x-2)}$ from $\frac{1}{x^{2}-2x}$.
Jun
10
comment Evaluating $\int \frac{dx}{x^2 - 2x} dx$
My intention is not to complain, if it seemed like that then I am glad I can correct the misunderstanding. I'm just having hard time seeing why this answer is useful. Do you really think that $x^{2}-2x=x(x-2)$ is what the OP is having hard time to figure out?
Jun
10
comment Evaluating $\int \frac{dx}{x^2 - 2x} dx$
You start from $\frac{1}{x^{2}-2x}$ and end up with $\frac{1}{x(x-2)}$ with 2 unnecessary middle-steps. You could just take $x$ as common multiplier in the first place.
Jun
10
comment Evaluating $\int \frac{dx}{x^2 - 2x} dx$
Why didn't you just take $x$ as a common multiplier? Why did you need all those middle steps?
Jun
10
comment convergence uniformly of a sequence of functions
I edited my answer according to the newly edited question, and as far as I'm concerned, the current claim does not hold.
Jun
10
revised convergence uniformly of a sequence of functions
added 698 characters in body
Jun
10
comment Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
@StefanWalter: I don't.
Jun
10
comment Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
@StefanWalter: To get back to the topic: each individual number on the real line has probability zero when throwing a dart, yet you always hit one of them. Once you did hit one, an event with probability zero occured. Being likely to happen and being possible to happen are two different things.