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Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

Real analysis is a branch of mathematical analysis, which deals with real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness, and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

145439 questions
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Infimum of a set

If $a$ is a positive number, show that $\inf \{a/n: n \in \mathbb{Z}^{+} \} = 0$. So let $A = \{a/n: n \in \mathbb{Z}^{+} \}$. Then $A$ is bounded below by $0$. Hence $\alpha = \inf(A)$ exists. So $0 \leq \alpha$. Now $\alpha \leq a/2n$ which…
Damien
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Is there a formula to figure this out?

I am nursing/pumping milk for my son and want to figure out how much longer I have to nurse/pump to be able to give him breast milk until he turns 1. I feel like this can be figured out via math but my brain isn't working well enough these days to…
Michelle
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If $f$ is monotonically increasing as well as monotonically decreasing on $I$, then show that $f$ is constant on $I$

Let $I$ be an interval and let $f$ : $I \rightarrow\mathbb{R}$ be a function. If $f$ is monotonically increasing as well as monotonically decreasing on $I$, then show that $f$ is constant on $I$. My approach: Consider $x,y\in I$ such that $x\neq y$.…
urt43as
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Subset of the reals that contains $x$ or $-x$ for every non-zero $x$, and is closed under finite addition

Is there a subset $A$ of the real numbers such that $(\forall \ x\in\mathbb{R}\setminus\{0\})$ exactly one of $x$ and $-x$ belongs to $A$; $A$ is closed under finite addition: $A+A\subseteq A$, or, in other words, for all $x_1, \ldots , x_n$ in $A$…
Arthur
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Equation has exactly one real solution

I am trying to show that the equation $$ 4x^2y^4+12x^2y^2+4x^2+4xy^2+4x+1=0 $$ has exactly one real solution $x,y$ and to determine it. The first observation is: If $y=0$, we are left with $$ 4x^2+4x+1=0 $$ which is solved by $x=-1/2$. Thus a real…
Rhjg
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Help Understanding the Heine-Borel Theorem Proof

I am struggling to understand the proof of the Heine-Borel theorem in "Measure, Integration and Real Analysis" by Sheldon Axler. The Theorem states: Every open cover of a closed bounded subset of $\mathbb{R}$ has a finite subcover. The part of the…
user12345
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How to prove $X = \{x\in \mathbb{R} \ : \ x^2 < a \} $ has supremum?

Assume that $a>0$, Suppose we have : $$X = \{x\in \mathbb{R} \ : \ x^2 < a \}$$ We should prove that this set has a supremum, and that's $\sqrt{a}$ . I saw this answer on one of the related posts: Suppose that $a>0$ then $\sqrt{a}$ is an upper…
Arian Ghasemi
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Disproving that $\sqrt{4}$ is irrational with the same logic proving that $\sqrt{2}$ is irrational

I have looked up another question regarding this disproof, but I got confused. If I understood it correctly, the disproof flows like this: Just like we have shown that $\sqrt{2}$ is irrational by contradiction, 'Assuming' that $\sqrt{4}$ is…
patha
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Showing range is countable

Let $f: \mathbb{R} \to \mathbb{R}$. For every $x \in \mathbb{R}$, there exists $\delta$, for every $y \in N(x, \delta)$ ($N$ stands for neighborhood) $f(y) \geq f(x)$. Show that the range of $f$ is countable.
Leitingok
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An application of Fatou's lemma and a differential function

Let $f\colon \mathbb{R} \to \mathbb{R}$ be a differentiable function (the derivative is not always continuous). Let $f$ satisfy the following condition. \begin{equation} \sup_{n \in \mathbb{N}} \int_{-1}^1…
sharpe
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A function that is continuous at the origin but not differentiable there

Show that $f(x,y)=\dfrac{xy^2}{x^2+y^2}$ (with $(x,y)\not=(0,0)$ and $f(0,0)=0$) is continuous but not differentiable at $(0,0)$. I tried to show continuity with an $\epsilon -\delta$ argument but I don't know how to factorize the expression so that…
Xena
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Proving that a function is bounded

This is one part of an exercise in my homework, which for some reason I can't think of any way to prove. $\displaystyle f(x,y)=\frac{xy^2}{x^2+y^4}$, if $(x,y)\neq (0,0)$ and $0$ otherwise. I'm trying to prove that this function is bounded. I have…
Leonardo Fontoura
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Showing infimum of distance between sets is positive

I am trying to understand the proof of Lemma 45.1 in Bartle, The Elements of Real Analysis. The lemma is used to prove that diffeomorphisms map sets with zero content into sets with zero content. Bartle leaves to reader (with hint that is…
WoodWorker
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Equating Lebesgue and Riemann integrals

I'm trying to work my way through Bartle's Elements of Integration and I am currently in Chapter 4, entitled the integral. If $f$ is a nonnegative measurable function, then its integral is defined as $$\int f\,d\mu=\text{sup}\left\{\int \phi\,d\mu:\…
David
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$q$ be a real polynomial of real variable $x$ of the form $q(x)=x^n+a_{n-1}x^{n-1}+....+a_1x-1 .\,\,$

I am stuck on the following problem: Let $q$ be a real polynomial of real variable $x$ of the form $q(x)=x^n+a_{n-1}x^{n-1}+....+a_1x-1 .\,\,$ Suppose $q$ has no roots in the open unit disc and $q(-1)=0.$ Then which of the following options are…
learner
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