Is the Axiom of Dependent Choice necessary in this proof?
While typically, the Axiom of Choice and its peripheral arguments are not emphasized in one's first exposure to Real Analysis, I am trying to be as rigorous as possible in my learning as an axiomatic...
View ArticleEquivalent definitions of affine function
For a function $f\colon\mathbb{R}^n\to\mathbb{R}$ the following two are equivalent\begin{align*}&\text{(i) there exist $a\in\mathbb{R}^n$, $b\in\mathbb{R}$ so that...
View ArticleProve that $\text{glb}(A)=\text{lub}(A) \iff A$ contains just one element.
Prove that $\text{glb}(A)=\text{lub}(A) \iff A$ contains just one element.I understand that if a set has only one element, say $x$, then $x$ would be the lower upper bound as well as the greatest lower...
View ArticleRelation between the second derivative and the second degree polynomial...
Let E $\subseteq \mathbb{R}$, $a$ be a limit point of $E$, $c \in \mathbb{R}$ and $f, h: E \to \mathbb{R}$ be differentiable functions such that $f(x) = f(a) + f'(a)(x - a) + c(x - a)^2 + h(x)$ for all...
View ArticleWhy can't $\mathbb{R}/\mathbb{Q}$ be linearly-ordered without Axiom of Choice?
This Question has an answer which is the only source that I can find about how $\mathbb{R}/\mathbb{Q}$ cannot be linearly ordered. I couldn't manage to open either of the source links provided in the...
View Article$n(fx)=f(nx)$ for natural number $n$ and $f(x)$ is an increasing real...
It is known that, for real increasing function $f:[0,1]\to\mathbb R$, if $f(ax)=af(x)$, then $f$is linear.Now consider $a=n$ where $n$ is natural.Is it possible that $f(x)$ is not linear on a positive...
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Inequality involving exponentials and factorials
Let $b>1$ (a base), $n\ge 2$ and $1\le k\le n$. I would like to know for which $k$ the inequalities$$\frac{1}{n!}b^{n-1}\le \frac{b^k-1}{(n-k)!}\le\frac{1}{(n-1)!}(b^n-1)(b-1).$$hold or at least...
View ArticleIs a real line an elementary set?
Now I'm studying Measure party from Rudin's PMA.It says that, if a set $A$ can be written as a union of finite number of intervals, then $A$ is an elementary set.(Now, I will denote the colletion of...
View ArticleWhy is $\partial A=(\overline{A}\setminus A)\cup (A\setminus A^\circ)$, where...
My book on analysis is using the following definitions (in $\mathbb{R}^k$):$$A^\circ=\{x\in \mathbb{R}^k\mid \exists r>0:B(x,r)\subseteq A\}\tag{1}$$$$\overline{A}=\{x\in \mathbb{R}^k\mid \forall...
View ArticleGood exercise selection from "Understanding Analysis", 2nd edition by Stephen...
I want to learn real analysis using the textbook "Understanding Analysis", 2nd Edition, by Stephen Abbott. My issue is, however, that I can't decide which exercises to do. Doing all of them is not an...
View ArticleHow did they arrive at the inequality [closed]
How did they conclude the inequality just after 25 which is below the word hence
View ArticleRudin Chapter 1, Exercise 20: new identity element for Dedekind cuts
I am working on exercise 20 in Chapter 1 of Baby Rudin on Dedekind cuts. With reference to the Appendix, suppose that property (III) were omitted from the definition of a cut. Keep the same definitions...
View ArticleIf $f$ is log-convex then $f$ is convex
Here's my attempt:$f$ is log-convex. Then:$\log f(\lambda x + (1-\lambda)y )\leq \lambda \log f(x) + (1-\lambda) \log f(y)$As $e^x$ is increasing, we can apply it to the inequation without changing its...
View ArticleInequality $\forall a,b\in\mathbb{R}_{*}^{+}~~\text{then}~~...
$\forall a,b\in\mathbb{R}_{*}^{+}\text{ then }\frac{1}{a}+\frac 1b+ab\geq 3$.Now this inequality: $a+\frac {1}{a}\geq 2$; $a+b\geq 2\sqrt{ab}$.But I can't use it!
View Articlepow and its relative error
Investigating the floating-point implementation of the $\operatorname{pow}(x,b)=x^b$ with $x,b\in\Bbb R$ in some library implementations, I found that some pow implementations are inexact.The...
View ArticleInjective monotonic mapping from rationals $\mathbb Q^2$ to $\mathbb R$
Exercise: $f: \mathbb Q^2\to\mathbb R$. Where $\mathbb Q$ is the set of rationalnumbers.$f$ is strictly increasing in botharguments.Can $f$ be one-to-one?This question is related to many questions...
View ArticleQuestion About Signed Measures
I am self-studying signed measure, and I come across the following construction:Let $\mu$ be a signed measure on the measurable space $(X,\mathscr{A})$, and let $A$ be a subset of $X$ that belongs to...
View ArticleBaby Rudin Chapter 1 Exercise 20: Disproving the additive inverse
In Baby Rudin Chapter 1 Exercise 20, we are asked to show that the additive inverse property fails if we omit property III of "cuts" i.e. that a cut $A$ has no largest element.For those unfamiliar, a...
View ArticleIs there a function $f:\mathbb{R}\rightarrow\mathbb{R}$ that maps every...
Functions like Conway's base-13 function map each open set to the whole real line. Functions like this one I think map every set of positive Lebesgue measure to the real line but I don't know how to...
View ArticleFinding a bound for optimal solution of a quadratic optimization problem
Suppose that $A\succeq 0,$$0\le y$ and $\|y\|\le \|x_\rho\|, \mu \in \mathbb R^n,$ and $e$ be the vector of ones. My goal is to show $\{x_\rho\}$ is bounded when large enough $\rho$ goes to $+\infty$,...
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