Doubt in countability of a subset of real number
Let $S=\{x \in \mathbb{R} \mid x>1, \frac{1-x^4}{1-x^3}>22\}$ then $S $ isEmptyCountably finiteCountably infiniteUncountable.My approach is:$\frac{1-x^4}{1-x^3}>22, x>1$$\implies...
View Article$p\log p \prec H(p)p$: Any probability vector majorizes its associated...
I guess the following inequality$$\sum_{i=1}^n g (p_i) \ge \sum_{i=1}^n g \left (\frac{-p_i \log p_i}{H(\boldsymbol{p})} \right )$$holds for any continuous convex function $g$ and any probability...
View Article$f_n \rightarrow f$ a.e. and $(\mu(A) < \infty \implies \int_A f_n d\mu...
Question: If $f_n \rightarrow f$ almost everywhere and $\int_A f_n d\mu \rightarrow \int_A f d\mu$ whenever $\mu(A) < \infty$, for $\mu$ a $\sigma-$finite measure on $X$, is it true that $\int_X f_n...
View Article$\sum_{n=0}^{+\infty} \frac{x^n}{1+nx} \stackrel{x \to 1^-}{\sim} -\ln(1-x)$
How to prove that $\sum_{n=0}^{+\infty} \frac{x^n}{1+nx} \stackrel{x \to 1^-}{\sim} -\ln(1-x)$ ?First try : I tried to compare the sum with the integral $\int_0^{+\infty} \frac{x^t}{1+tx}\mathrm{d}t$....
View ArticleCan difference quotient sets be nowhere dense?
Let $f:\mathbb{R} \to \mathbb{R}$ be a function and consider the difference quotient set $$D_f = \left\{\frac{f(y) - f(x)}{y-x} : (x,y) \in \mathbb{R}^2, y > x\right\}$$Can $D_f$ be nowhere dense in...
View ArticleExistence of the Riemann Stieltjes integral of a discontinuous function with...
It have read multiple times online and on this forum that when $f:I\to \mathbb R$ and $\alpha :X\to \mathbb R$, with $I\subseteq X$ and $X$ is a closed interval, if $f$ and $\alpha$ are discontinuous...
View ArticleBernoulli shift on $S^\mathbb{Z}$
Why is the Bernoulli shift ergodic? I know the proof for $S^\mathbb{N}$. The best would be if you could argue from this fact to the integer case, so that way I can see how one usually goes from...
View ArticleFind a function in the unit sphere of a function subspace with given properties
Preliminaries.$X$ is a compact Hausdorff space.$a$ is a fixed element of $X$.$C(X)$ denotes the space of real-valued continuous functions on $X$$A=\{f \in C(X); f(a)=0\}$ is a Banach space respect to...
View ArticleWhat's an example of a discontinuous linear functional from $\ell^2$ to...
I'm trying to find a discontinuous linear functional into $\mathbb{R}$ as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since $\ell_2$ is infinite-dimensional,...
View ArticleUse Cut Property to prove Axiom of Completeness
I've already proved the Cut Property (CP) using the Axiom of Completeness (AoC), but want to now show the converse. Here are my book's definitions:Cut Property: If $A, B$ are nonempty, disjoint sets...
View ArticleEvaluate $\lim_{n \rightarrow \infty} \int_0^{\infty} ne^{-nx}...
I'm trying to evaluate $\lim_{n \rightarrow \infty} \int_0^{\infty} ne^{-nx} \frac{x^2+1}{x^2+x+1 dx}$. I came across this post detailing a solution to it, and wanted to ask if my solution is...
View Articleshow that, the function f is injective
Consider $f : [0, 1] \to \mathbb R$ defined as:$$f(x) = \begin{cases}x &: x \text{ irrational} \\x + \sqrt{2} &: x \text{ rational}\end{cases}$$I need to show that, $f$ is injective.My...
View ArticleShowing that $\sum_{a\in A} 4^{-a}\neq \sum_{b\in B}4^{-b}$ where $A\neq B...
Let $A,B\in P(\mathbb{N})$ with $A\neq B$. Note that $A, B$ can be finite or infinite. Show that$$\sum_{a\in A} 4^{-a}\neq \sum_{b\in B}4^{-b}$$So I'm not entirely sure this claim is true, but I think...
View ArticleImage of an increasing function with discontinuities on a dense countable...
When discussing this question, we made interesting observations on increasing (i.e., nondecreasing) functions with discontinuities on rationals. Let $f: [0,1] \to [0,1]$ be an increasing...
View ArticleWhy is the derivative operator $D:C^\infty([0,1])\to C([0,1])$, $Df\equiv...
This is an example I read around closed graph theorem. Let $Y=C[0, 1]$ and $X$ be its subset $C^\infty[0, 1]$. Equip both with uniform norm. Define $D: X \to Y$ by $f \mapsto f'$. Suppose $(f_n, f_n')...
View ArticleDoes $\sum_{k\geq0}\frac{z^k}{k!^s}$ only have negative real roots?
It might be complicated to decide whether or not a given entire function has only real zeros. So, I'll focus on special examples. It is known that the following function given by the infinite...
View ArticleFormalising proof of a rational number between any two reals number
I feel my question is a symptom of a larger issue I have with analysis, which is being unable to get the exact proof even when you have the right idea. It's also an issue of knowing how much rigour is...
View ArticleProof that $e=\lim_{n\rightarrow\infty}\left(1+\frac1n\right)^n$ [closed]
From the definition of $e$ as the number such that $\lim_{h\rightarrow0}\frac{e^h-1}{h}=1$, is it possible to derive the formula $e=\lim_{n\rightarrow\infty}\left(1+\frac1n\right)^n$ withtout using any...
View ArticleAre all points $x$ with $f'(x) = 0$ local maxima, local minima, or stationary...
In this post, I am considering only functions of the form $f:\mathbb{R}\rightarrow\mathbb{R}$. It's common to classify stationary points into local maximum points, local minimum points, and saddle...
View ArticleIs an infinite composition of bijections always a bijection?
Main QuestionSuppose I have a sequence of real valued functions $f_1:X_0\rightarrow X_1,...,f_n:X_{n-1} \rightarrow X_n,...,$ and I then, with $\circ$ denoting function composition, define$$g_n : X_0...
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