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Doing math.


Jan
31
comment Show that sequence converges pointwise to a function that is not Riemann Integrable.
$C^{0}([a,b])$ might denote continuous functions $[0,1]\to \mathbb{R}$.
Jan
31
comment 'uniform approximation' of real in $[0,1]$
Are $n$ and $N$ the same number here? Also, what is $\varepsilon$?
Jan
29
comment How can a bounded subspace of the left order topology be compact?
Why is $(0,1)$ bounded? In other words, what notion of boundedness do you use? Especially since we are not looking at the standard topology on $\mathbb{R}$ with the standard metric to measure distances.
Jan
15
comment Convolution of integrable function with bounded function
What is the relationship of $H$ and $K$? Are they denoting the same function?
Jan
9
comment If $P(E_n) = 0$, then $P(\cup E_n) =0$
you probably have a typo there when using subadditivity: there shouldn't be a union after $\leq$, just the probabilities of the sets $E_{n}$.
Jan
6
awarded  Yearling
Jan
2
answered $G_\delta$ sets
Dec
20
comment Definition of functions on metric spaces.
Why would that cause any trouble? And which other way would you prefer to use to define a function?
Dec
8
comment Are there any measurable, uncountable sets dense in [0,1] with Lesbegue measure less than 1?
@user70147: You could do something like this. Enumerate the rationals in $[0,1]$ by $\{q_{k}:k\in\mathbb{N}\}$, and take $\frac{1}{2^{k+2}}$ radii ball around each $q_{k}$. Take the union of these balls intersected with $[0,1]$. It has measure at most $\frac{1}{2}$ and bigger than $0$, it is dense, and what can you say about its intersection with every interval on $[0,1]$? (since every interval contains rationals)
Dec
8
answered Are there any measurable, uncountable sets dense in [0,1] with Lesbegue measure less than 1?
Dec
8
comment Proof for Image of Indexed Collection of Sets?
@Randomstop. You're welcome. By the way. If you found this answer useful, and also in case for future answers, you can always up vote them by clicking the "arrow up" next to it and click the "accept" mark below the score meter.
Dec
6
comment Proof for Image of Indexed Collection of Sets?
@Randomstop. Precisely. Except for the first line you probably meant $x\in U_{\alpha}$ for all $\alpha$ instead of $x\in \alpha$ for all $\alpha$. Otherwise, what you wrote is exactly what you wanted to conclude.
Dec
6
comment Proof for Image of Indexed Collection of Sets?
@Randomstop. Ideally you would want that $y\in f(\bigcap_{\alpha\in\Lambda} U_{\alpha})$. But you already know that $y=f(x)$ where $x$ is a member of all $U_{\alpha}$ (by considering $x=x_{\alpha}=x_{\beta}=..$). So you're practically almost there.
Dec
6
comment Proof for Image of Indexed Collection of Sets?
@Randomstop. Yep, you got it.
Dec
6
comment Proof for Image of Indexed Collection of Sets?
@Randomstop. Exactly. So in this scenario you know that $f(x_{\alpha})=y$ for all $\alpha$. So, if you take any indices $\alpha,\beta$, what do you know of $f(x_{\alpha})$ and $f(x_{\beta})$? And what does that tell you about the relationship of $x_{\alpha}$ and $x_{\beta}$?
Dec
6
comment Prove that $\lim_{(x,y)\to(0,0)} \frac{x^3y}{x^6+y^2} = 0$
Is $\frac{x^{3}y}{x^{6}+y^{2}}$ always non-negative? Even for points arbitrarily close to the origin?
Dec
6
comment Proof for Image of Indexed Collection of Sets?
@Randomstop: Since $x_{\alpha}$ is in the domain and $y$ in the range, we can't necessarily say that $x_{\alpha}=y$. But does the injectivity of $f$ imply something about the relationship of each $x_{\alpha}$ to each other?
Dec
6
revised Cardinalities of power sets, $\mathbb{N}$, and $\mathbb{R}$.
tex edits
Dec
6
revised for any $A\subseteq X$, $f(\overline{A})\subseteq\overline{f(A)}$ , if and only if $f: X \to Y$ is continuous.
tex edits
Dec
6
revised Finding a continuous function $f:(X,T_u)\to (X,T_e)$ with $f(0)=0$ and $f(1)=1$ that has some $c\in (0,1)$ such that $c\notin f(X)$.
some tex edits