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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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fundamental theorem of calculus relation

I sent this question last week but it turns out that I had an error in the formula that I was trying to understand. So, here goes a second try. I am reading a proof in a convex optimization book by Nesterov and at the beginning of the proof it says…
mark leeds
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Determine if the set is compact

$$S=\{1,1/2, 2/3,3/4,....\}$$ I think this is compact as it has one sequence that covers all elements in set except $1$. This sequence is $a_n=\frac n{n+1}$. This sequence converges to $1$ hence all subsequences in $S$ converge to $1$, which is in…
Sonal_sqrt
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For any non-empty bounded below set $A$ such that $A \subset \mathbb{R}$, show that $-\inf(A)=\sup(-A)$.

There are various proofs of this on MSE but I didn't think any of them was quite rigorous enough. I've already proved that if $A \subset \mathbb{R}$ is non-empty and bounded below, then it has an infimum. $A$ is bounded below and non-empty, so…
user401936
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Does $\sum_{n=1}^{\infty}(\sin{n})^n$ converge?

As above, does $\sum_{n=1}^{\infty}(\sin{n})^n$ converge? And if so, to what value. From calculating partial sums, it appears it might, but I'm not quite sure how to proceed from there.
David Reak
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Weakly convex function that is not convex?

I’m reading Emil Artin’s introduction on the Gamma function, he proved in Theorem 1.5 that “a function is convex, if and only if, it is continuous and weakly convex”. The definition Artin used is: The difference quotient: …
athos
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It is true that $\sum_{1 \leq n, m \leq N} \cos(a_n - a_m) \geq 0.$

I want to prove or find a counterexample of the following proposition: Let $N$ be a positive integer and $a_1,\dotsc,a_N$ be distinct real numbers. Then it holds that: $$\sum_{1 \leq n, m \leq N} \cos(a_n - a_m) \geq 0.$$ For $N=1,2$ the result…
Gabriel
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Definition of an algebra on a set

Definition Let $X$ be a set and $\mathcal{E} \subset \mathcal{P}(X)$. The family $\mathcal{E}$ is elemental if: $\emptyset \in \mathcal{E}$; $E \cap F \in \mathcal{E}$ for each $E , F \in \mathcal{E}$; For all $E \in \mathcal{E}$, there is a finite…
joseabp91
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Prove that $f(x)=\sqrt[n]{x}$ is not Lipschitz continuous

Show that none of the functions $f(x)=\sqrt[n]{x}$, for $n\in \Bbb N$, $n\ge2$ are Lipschitz continuous. So i did it this…
Max
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If $\|x_{n+1}-x_n\|\to 0$ and $\|x_n\|$ is bounded, then $x_n$ converges.

Is there a quick and easy way of proving that if $\|x_{n+1} - x_n\|\to 0$ and $\|x_n\|< B$ then $x_n$ converges? If $\|x_n\|< B$, then there is $x_{n_k}\to x^*$ (by Bolzano Weierstrass). Then I thought maybe I could write $$ \|x_n - x^*\| < \|x_n -…
user3002473
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Show that $\sum_{k=1}^\infty a_k<\infty \implies \sum_{k=1}^\infty a_k^3<\infty $

Let $(a_k)$ a sequence of real number. How can I show that $$\sum_{k=1}^\infty a_k<\infty \implies \sum_{k=1}^\infty a_k^3<\infty.$$ If $(a_k)$ has constant sign, it's obvious. But unfortunatly it's not supposed, so comparaison criterion doesn't…
user352653
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How to convert $\int_{0}^{\infty} \sin (t^2) dt$ to a limit of a series?

Possible Duplicate: Prove: $\int_{0}^{\infty} \sin (x^2) dx$ converges. I have shown that; $\forall \epsilon>0, \exists r\in \mathbb{R}$ such that $ \forall x,y>0, r
Katlus
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Is $\sin(2x)=x$ solvable?

($x \in \mathbb{R}$) Graphically, it's obvious that the equation should have 3 solutions for x, but I can't think of any way to solve this without resorting to computation of [the Maclaurin series for $\sin(2x)$]$\div x$ or some cleverer…
Meow
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Why does the Ratio Test prove that this particular sequence converges on 0?

I was recently reading How to Think about Analysis by Lara Alcock in preparation for taking Real Analysis in the winter, and ran into the following theorem regarding sequences that either tend towards infinity or converge on a particular number,…
EJoshuaS - Stand with Ukraine
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a question on orthonormal basis

Suppose $\{u_n\}_{n=1}^{\infty}$ is an orhtonormal basis in $L^2[0,1]$, prove that $\sum_{n=1}^{\infty}|u_n(x)|^2=\infty$ for almost every $x\in [0,1]$. Any hint on this problem? I tried to prove the set $Y$ has measure zero, where $Y=\{x\in [0,1]:…
ougao
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Multiples of a given irrational number can be arbitrarily close to a natural number

How do I prove the following: For every irrational number $q$, given $\varepsilon>0$, there exist natural numbers $N$ and $M$ such that $|Nq-M|<\varepsilon$.
user7741
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