# Tag Info

### Twice differentiable Lipschitz functions have Lipschitz gradient

Consider $f(x) = \int_{0}^{x}\sin(t^2)\,dt$. Note $f'(x) = \sin(x^2)$ is bounded, while $f''(x) = 2\cos(x^2)x$ is unbounded, so $f$ is Lipschitz and $f'$ is not Lipschitz.
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### How can I show that $\bigcap\bigcup_{k=0}^{n-1} [{k \over n}, {k+1 \over n}]^2=\{(x,x):x\in [0,1]\}$

I use $\langle a,b\rangle$; $a,b \in \mathbb{R}$; to denote the point in $\mathbb{R}^2$ with first coordinate $a$, second coordinate $b$, I do this to not overload the notation $(a,b)$, which may be ...
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### Discretized Distributions on Rationals?

I would say that "nice", "benchmark" continuous disributions tend to have regular supports - closure of the set on which its density (w.r.t. the Lebesgue measure) is positive. E.g. ...
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### Discretized Distributions on Rationals?

Expanded from comments: I do not see how you plan to sum (as opposed to integrate) a density (as opposed to a probability mass function) and get 1. You could create an discrete distribution on the ...
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### Does this slight modification alter the limit of this sequence?

Taking a logarithm and applying l'Hopital gives \begin{align} y &=\lim_{n\to\infty} \left[1- \frac{x^2}{n}\left(1+\frac{2}{3}\frac{x}{\sqrt{3n}}\right)\right]^{n/2} \\ \ln y &= \ln\lim_{n\to\...
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### How to give this sum a bound?

Expanding on the idea from my comment and the attempt by the OP, I will establish $8\times 15 \times \left(\frac{\pi^4}{90}\right)^2$ is an upper bound for the sum, assuming $x,y\neq0$ in the sum. ...
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### Real Analysis Question about Limit points and ε-neighborhoods

To prove the statement If every $\varepsilon$-neighborhood of $x$ intersects $A$ at some point other than $x$, then $x$ is a limit point of $A$. where your definition has $x$ is a limit point of $A$...
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One way to circumvent this problem would be to take the $s_1 = s$ and $s_2 = t$ (you went from one notation to another, that is something to look out for) corresponding to $\varepsilon/2$ instead of \... • 5,849 0 votes ### Newton approximation in Tao Analysis 1 $$|f(x) - (f(x_0) + L(x - x_0))| = |f(x) - (f(x_0) + f(x) - f(x_0))|$$ Why would this be the case? Just because we have L = \lim_{\substack{x \to x_0 \\ x \in X \\ x \ne x_0}} \frac{f(x) - f(... • 45.9k 1 vote Accepted ### A= \{(x,y,z) \in \mathbb{R}^3:x^2+2y^2+z^2 < 4z\} limit: \lim_{n \to \infty}\frac{1}{n} \int_A \frac{y^2z}{ln(x^2+2y^2+n) - ln(n)} \ d\lambda_3 Sketch: On A, x^2+2y^2<4z-z^2. The LHS is non-negative, so the RHS must be non-negative which gives z \in [0,4]. Thus, \begin{align*} A &\subset B := \{(x,y,z): z \in [0,4], x^2+2y^2<\... • 91 4 votes Accepted ### Translates of a set of positive Lebesgue measure cover \mathbb{R}? Let E be fat Cantor set (i.e. a Cantor like set of positive measure). Then E has no interior. If \mathbb R=\bigcup_n (E+x_n) then Baire Category Theorem implies that E+x_n has an intetior ... • 39.2k 0 votes ### Infinite Summation of Almost Sure Convergent RVs Note that by setting X_{n,i} = 0 for i>b_n the statement is equivalent to \sum_{i=1}^{\infty} X_{n,i} \to \sum_{i=1}^\infty X_i almost-surely as n\to \infty. In turn, by considering the (... • 321 0 votes Accepted ### How to show that the limit of a sequence is not equal to some value? I will clarify my comment. By definition, l is the limit of the sequence a_n if for every \varepsilon>0 you can find N such that \left|a_n-l\right|<\varepsilon for all n>N. Thus, ... • 194 2 votes Accepted ### weak convergence and pointwise implies L_p convergence In general, the statement in the title of the OP is false. In ((0,\infty),\mathscr{B}(\mathbb{R}),m), m is Lebesgue measure, define f_n(x)=\frac1{x-n}\mathbb{1}_{(1,\infty)}(x-n). f_n\... • 40.8k 1 vote Accepted ### Approximation of a class of measurable functions by simple functions with "compact domain" This depends on precisely what you meant by "approximated by". If you meant the exact same thing as in the first paragraph, i.e., there exists a sequence S_n = \sum_{i = 1}^{k_n} c_{(n, i)}... • 9,953 0 votes Accepted ### The radius of convergence using root test The root test determines the radius of convergence. For the series \sum_{k=0}^\infty a_k z^k, we have radius of convergence R, where \frac{1}{R} = \limsup_{k \to \infty} |a_k|^{1/k} . $$This ... • 113k 0 votes Accepted ### How do I Show that \exists a, \forall e > 0; a < e \Rightarrow a \le 0 What you want to show is: if a constant a is smaller than any arbitrary positive number, then it must be non-positive, that is, a \leq 0. The problem is that the statement in logical notation in ... 2 votes ### Constructing the interval [0, 1) via inverse powers of 2 There are already good answers, but just to show that your intuition could be followed to the end. This answer is not completely rigorous and it's certainly more work than just the simple ... • 5,875 2 votes ### How do I Show that \exists a, \forall e > 0; a < e \Rightarrow a \le 0 Be careful when using both existential and universal quantifiers as you can’t swap their order ! For example your first statement is “for all a, it exists some e such that blabla”. In this ... • 395 3 votes Accepted ### Continuity and lebesgue integrability of integral function, proof verification This seems to be a correct proof. For the convergence \chi_{E_n}\rightarrow\chi_E, consider points t in \mathbb{R} different from from x+h and x-h; then there is a positive \delta for ... 2 votes ### Continuity and lebesgue integrability of integral function, proof verification I will asume that h>0. The case h<0 is left to you.$$2h\|\phi_1\|=\int_{-\infty}^{\infty} |\int_{x-h}^{x+h} f(t) dt| dx \le \int_{-\infty}^{\infty} \int_{t-h}^{t+h} |f(t)| dx dt=\... • 39.2k 4 votes ### Constructing the interval [0, 1) via inverse powers of 2 You made an interesting query in the comments: \def\lfrac#1#2{{\large\frac{#1}{#2}}} $I was thinking of something like$x = \lfrac{a_1}{2}+\lfrac{b_1}{3}+\lfrac{c_1}{5}+\cdots+\lfrac{a_n}{2^n}+\...
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