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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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decreasing sequence of nonempty closed sets in M

Let $( M , d )$ be compact. Suppose that $( F_n)$ is a decreasing sequence of nonempty closed sets in $M$, and that $ \bigcap_{n=1}^{\infty} F_n$ is contained in some open set $G$. Show that $F_n \subset G$ for all but finitely many n . I know how…
Hamilton
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Rational contained in open interval centered at an irrational.

Lemma. Let $y \in [0,1] \setminus \mathbb{Q}$, $\epsilon > 0$ and $N \in \mathbb{N}$ such that $\frac{1}{N} < \epsilon$. Then one can choose $\delta > 0$ small enough such that $$ \frac{p}{q} \in (y - \delta, y + \delta) \Rightarrow q > N,…
IsaacR24
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Prove that if $f$ and $g$ are integrable functions on $[a, b]$ such that $f(x) = g(x)$ almost everywhere, then $\int^b_a f = \int^b_a g$

Say that $f\in\mathcal{R}[a,b]$ and $f(x)≥0$ almost everywhere. Say also that $\int^b_a f\geq 0$ I'm trying to prove that if $f$ and $g$ are integrable functions on $[a, b]$ such that $f(x) = g(x)$ almost everywhere, then $$\int^b_a f = \int^b_a…
Bob Mcdonald
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Approaching analysis with a discrete mindset

This question is slightly subjective, so I'm not sure this is the right place to ask it. Algebra and analysis are the meat and potatoes of modern mathematics. I have good intuition for discrete math, so I tend to lean on shaky metaphors to think…
yberman
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a problem on roots of a real polynomial $p$ that has no roots in the open unit disc and $p(-1)=0$

Let $p$ be a real polynomial of the real variable $x$ of the form $$p(x)=x^n+a_{n-1}x^{n-1}+...+a_1x-1$$Suppose that $p$ has no roots in the open unit disc and $p(-1)=0$.Then which is/are true? $1$.$p(1)=0$. $2$.$\displaystyle{\lim_{x\to…
poton
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Area under the function $f$

Let consider the function $f:\mathbb R \rightarrow I$, where $I$ is $(0,1)$. Consider that $f$ is both continuous and bijective. Let's assume that it is strictly increasing. Now consider a real number $\alpha \in I$. What do we have to prove is…
shangq_tou
  • 998
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How can I prove that a limit of a function exists if and only if both limit sides exist and are equal?

I understand the concept behind this and it is quite obvious to me as to why but I am having problems proving this rigorously. To make things more "simple," imagine that there is function $f: [-2,2] \rightarrow \mathbb{R}$ and I must prove that…
TreausreDragon
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Why is $\sum_{n=1}^\infty \frac{1/n!}{x^2 + 1/n^2}$ not analytic?

I've been told that the (real-valued) function $$f(x) = \sum_{n=1}^\infty \frac{1/n!}{x^2 + 1/n^2}$$ is "obviously" not analytic at $x=0$. Can someone help me see the reason? First, I verified that the series does converge for all $x$. I tried to…
Kelley Alan
  • 81
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Prove this inequality by using the mean value theorem

I want to prove that $x<\frac{2x}{2-x}, \forall x \in (0,1)$, by using the mean value theorem. So, consider $f(x)=\frac{2x}{2-x} -x$. $f(0)=0$. $f´(x)=\frac{2x-2}{(2-x)^2} - 1$ and $f'(x)<0, \forall x \in (0,1)$. By the mean value theorem: $$\exists…
Walter r
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Proof that if a function has a limit for large x, then the function is bounded.

I need help in order to confirm whether my proof is approved or not. It follows as: Claim: Let $f: [a,\infty) \mapsto \mathbb{R}$ where $f$ is continous. If $\exists \lim_{x \rightarrow\infty}f(x)=L$ for some $L\in \mathbb{R}$, then the function $f$…
Tanamas
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Baby Rudin Theorem 3.20 on sequences

Part (b) of the theorem states that "If $p>0$, then $\lim_{n\to\infty}p^{1/n}=1.$" Here is the $p\ge1$ case that Rudin proves: If $p\ge1$, put $x_n=p^{1/n}-1\ge0.$ By the binomial theorem we have $1+nx_n\le(1+x_n)^n=p.$ We then have$$0\le x_n\le…
Philip Shen
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How to understand $\delta$ and $\varepsilon$ in real analysis?

I'm a bit confused as to how to interpret $\delta$ and $\varepsilon$ mean in real analysis. My textbook gives an example demonstrating that $\frac{1}{x^2}$ is not uniformly continuous on $(0,1)$. Definition of not uniformly continuous: $(\exists…
Student
  • 3,353
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A general expression for the $n$th derivative

Can someone help me prove the following $$f^{(n)}(x_0)=\lim_{h \rightarrow 0}\frac{1}{h^{n}}\sum\limits_{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)(-1)^{n-k}f(x_0+kh) $$ I have managed to manipulate the above expression with l'Hôpital's…
user82888
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Why is this function bounded and Lipschitz?

Suppose $f \in \mathbb{R}[x]$ and define $g \colon \mathbb{R} \to \mathbb{R}$ by $$g(x) = \frac{f(x)^2}{(x^2+1)^{d+1}}, \text{where } d = \deg(f)$$ I'm looking for a quick proof as to why $g$ is bounded above and Lipschitz. Edit: $g$ is not proper…
James Rohal
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Stone-Weierstrass Theorem application

Suppose $f:[-1,1] \to \mathbb{R}$ is a continuous function such that $f(0) = f'(0) = 0$. I wish to show that for every $\epsilon > 0$ there exists a polynomial such that $$\sup_{x\in[-1,1]} |f(x) - x^2p(x)| < \epsilon $$ Letting $\epsilon > 0$, I've…
troutman314
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