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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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1-Variable Real Analysis True/False questions on Supremum, Infimum, and Inequalities with them

(S. Abbott. Understanding Analysis 1 ed. pp 18 question 1.3.9) is asking me to answer the following questions without any formal proofs. I have some intuition for them, but I was hoping to get some external input as well. It would be great if you…
confused
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Convergence of Sequences

I came across the following problems on convergence of sequences during the course of my self-study of real analysis: Suppose $a_n \to a$. Define $$s_n = \frac{1}{n}\sum_{k=1}^{n} a_k$$ Prove that $s_n \to a$. So $(a_n-a)$ is a null sequence. I…
Damien
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Sequences and Intervals

I came across another real analysis problem in my self study: Let $[a,b]$ be a closed interval in $\mathbb{R}$ and let $(x_n)$ be any sequence in $\mathbb{R}$. Prove that $[a,b]$ contains a real number not equal to any term of the sequence. I…
Damien
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proof of the continuity of as function

Let $f: \mathbb{R} \to \mathbb{R}$ be a surjective function such that for all non convergent sequences $(x_n)$, the sequence $(f(x_n))$ is non convergent. Prove that $f$ is continuous. Thank you
user62138
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Prove $(f+g)'(x) = f'(x) + g'(x)$

In Rudin's textbook, "Principles of Mathematical Analysis", theorem 5.3 says: If $f$ and $g$ are defined on $[a, b]$ and are differentiable at a point $x \in [a,b]$, then $$(f+g)'(x) = f'(x) + g'(x)$$ Rudin said this statement is clear by…
InfimumMaximum
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I need help interpreting Definition 1.10 in Rudin’s PMA.

While I have a basic understanding of what this definition states, I've been running into trouble interpreting the results when I test it with certain sets other than the traditional examples used to show that the Rationals do not have this…
CPmkI
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Continuous extension of a function

Can anybody help me with this problem? Justify whether the following statement is true or false: Every continuous function on $\Bbb Q\cap [0,1]$ can be extended to a continuous function on $[0,1]$. Any help will be appreciated.
Ester
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Proof that every cauchy sequence converges in $\mathbb R^k$

I'm having a hard time understanding this proof (the portion in bold). I know $E_N$ is bounded but how is the finite set $\{x_1, \ldots, x_{n-1}\}\,$ bounded? (Is it because every finite set in $\mathbb R^k$ is bounded?) I didn't get the last…
InfimumMaximum
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$f : \mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that for every fixed $y\in\mathbb{R}$, $f(x + y)-f (x)$ is a polynomial in $x$

Let $f : \mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that, for every fixed $y\in\mathbb{R}$, $f(x + y) - f (x)$ is a polynomial in $ x$. Prove that $f$ is a polynomial function. please give some hints for the problem. I tried to…
tuj8 saerty
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Absolute continuity on an open interval of the real line?

In classical real analysis I've only seen absolute continuity defined for functions on compact interval $[a,b]$, where the two equivalent definitions are: $f:[a,b]\rightarrow\mathbb{R}$ is AC if (1) Given $\epsilon > 0$ there is a $\delta > 0$ such…
Greg O.
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Showing a function is unbounded

Let $f(x)=x\cos x+\sin x$. Show that for every $M>0$ and every $k\geq 1$, there is $x_0>M$ such that $f(x)\geq k$, $\forall x\in [x_0,x_0+\frac{1}{k}]$. If we replace $x$ by $2\pi n$ then $|f(2\pi n)=|2\pi n|$. Can this help? Can I get a help to…
Unknown
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Open set as a countable union of open bounded intervals

Can every nonempty open set be written as a countable union of bounded open intervals of the form $(a_k,b_k)$, where $a_k$ and $b_k$ are real numbers (not $\pm\infty$)? If yes, can someone point me toward a proof? If not, counterexample? Note that…
David
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The continuity of measure

Let $m$ be the Lebesgue Measure. If $\{A_k\}_{k=1}^{\infty}$ is an ascending collection of measurable sets, then $$m\left(\cup_{k=1}^\infty A_k\right)=\lim_{k\to\infty}m(A_k).$$ Can someone share a story as to why this is called one of the…
David
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Hölder- continuous function

$f:I \rightarrow \mathbb R$ is said to be Hölder continuous if $\exists \alpha>0$ such that $|f(x)-f(y)| \leq M|x-y|^\alpha$, $ \forall x,y \in I$, $0<\alpha\leq1$. Prove that $f$ Hölder continuous $\Rightarrow$ $f$ uniformly continuous and if…
Walter r
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