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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. .

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Generating a $\sigma$-algebra from a $\sigma$-ring

Here is exercise 1.1(c) from Folland's Real Analysis: If $\mathcal{R}$ is a $\sigma$-ring, then $\mathcal{M}= \{ E\subset X : E \in \mathcal{R} \text{ or } E^c \in \mathcal{R} \} $ is a $\sigma$-algebra. Recall that a family of sets $\mathcal{R}…
user41894
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Prove that if $a \ge c$ for all $c < b$, then $a \geq b$

Let $a$ and $b$ be elements in an ordered field, prove that if $a \ge c$ for every $c$ such that $c \lt b$, then $a\ge b$. My proof idea below: Let $S = \{x | x
Huiwen Zheng
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Epsilon-Delta Proof for $x^n$ tends to 0

What is the epsilon proof that $x^n \rightarrow 0$ as $n \rightarrow \infty$ provided $|x| < 1 $? I only know it's true because I know the geometric series converges, which implies its terms must tend to 0, but never seen an epsilon proof of this…
Peter
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Existence of Limit of a Sequence in $\mathbb{R}$

Let $x_n$ be a bounded sequence such that $x_{n+1}\leq x_n + 1/n$ for all $n \in \mathbb{N}$. Then prove or disprove that $\{x_n\}_n$ always converges . I think that it is not necessarily convergent, but I could not manage to find a counter example…
Ester
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Can you replace $x$ with $x^2$ for any Maclaurin series?

Let's say I have a Maclaurin series for some function $f(x)$, and to find Maclaurin series $f(x^2)$ can I just substitute $x^2$ for each term Maclaurin series for $f(x)$?
user354021
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$\mathbb{R}$ is uncountable, Abbott's proof

There is this proof in Abbott's book (Understanding Analysis) on page 25, that I am failing to understand. Theorem is that $\mathbb{R}$ is uncountable. And here is how the author proceeds to prove it (I know there is an easier proof by Cantor): To…
Naz
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$f,g$ continuous functions, $f$ strictly increasing, non-vanishing: $\int_a^bg(x)(f(x))^n\,dx=0$ for all $n$ $\Rightarrow g\equiv 0$

While studying for an exam, I came across the following problem. Suppose $[a,b]\subset\mathbb{R}$ and $f:[a,b]\rightarrow\mathbb{R}$ is a strictly increasing non-vanishing continuous function. Suppose $g:[a,b]\rightarrow\mathbb{R}$ is a continuous…
John Adamski
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Baby Rudin Chapter 1 Exercise 9: Complex Field ordering

Problem: Suppose $z=a+bi$, $w=c+di$. Define $z
user249586
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Example of an open map on $\mathbb{R}^2$ that is not a submersion

I am searching for an example of a map $f : \mathbb{R}^2 \to \mathbb{R}$ that is open but is not a submersion... I know that any constant map is not a submersion, but it is indeed closed, I am wondering for an example where $f$ is an open map. I…
L.F. Cavenaghi
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Prove $f$ is not differentiable at $(0,0)$

For $$f(x,y)=\begin{cases} \frac{x|y|}{\sqrt{x^2+y^2}} & \text{ for }(x,y)\neq (0,0)\\ 0 & \text{ for } (x,y)=(0,0) \end{cases}$$ I'm trying to prove $f$ is not differentiable at $(0,0)$. I showed if $f$…
Kelan
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Proving every bounded infinite set has a limit point without using Bolzano-Weierstrass.

I am trying to prove that every bounded, infinite set has at least one limit point through the concept of open covers and without using Bolzano Weierstrass. I have no idea where to start. So far I've only assumed that a set A is bounded and…
tralaman
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For differentiable functions $f,g$, $\nabla f(x)=g(x)x$. Then $f$ is constant on S.

Problem saying that : $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is differentiable. Assume that there is a differentiable function $g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ such that $\nabla f(x)=g(x)x$ . Show that $f$ is constant on …
Darae-Uri
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to show $f(t)=g(t)$ for some $t\in [0,1]$

Let $f,g:[0,1] \rightarrow \mathbb{R}$ be non-negative, continuous functions so that $$\sup_{x \in [0,1]} f(x)= \sup_{x \in [0,1]} g(x).$$ We need To show $f(t)=g(t)$ for some $t\in [0,1].$ Thank you for help.
Myshkin
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integration of a continuous function $f(x) $ and $xf(x)$ is zero

Possible Duplicate: Prove that $\exists a
Myshkin
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Construction of $\Bbb R$ from $\Bbb Q$

As it is true that we can construct all rational numbers $\Bbb Q$ from the set of integers $\Bbb Z$, is it possible to construct the set of real numbers $\Bbb R$ from $\Bbb Q$? If yes, how? Is there any procedure? And if no, is there any proof that…
Aang
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