Mathematics Stack Exchange
Mathematics Stack Exchange

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
7
votes
1 answer

Does Rudin's definition of "bounded" overlook the empty set?

In Definition $2.18$ of Baby Rudin, Rudin defines boundedness for metric spaces as follows: given a metric space $X$, and a set $E \subset X$, we say $E$ is bounded if there exists a real number $M$ and a point $q \in X$ such that $d(p,q) < M$ for…
MathematicsStudent1122
  • 12,462
7
votes
8 answers

Prove that the equation has only one real root.

Prove that $(x-1)^3+(x-2)^3+(x-3)^3+(x-4)^3=0$ has only one real root. It's easy to show that the equation has a real root using Rolle's theorem. But how to show that the real root is unique? By Descartes' rule of sign, it can be shown that it has 3…
Jitu Biswas
  • 143
  • 1
  • 9
7
votes
1 answer

Composition of uniformly continuous maps

Fix $p>1$. Let $\mathscr C[0,2]$ be the space of continuous functions on $[0,2]$ with metric given by $$d_{p}(f,g)=||f-g||_p:=\left(\int_0^2 |f(x)-g(x)|^p \ dx\right)^{\frac{1}{p}}.$$ Consider the map $\Psi:\mathscr C[0,2]\rightarrow \mathbb R$…
Timothy
  • 533
7
votes
1 answer

Proving following holds for almost everywhere

I am studying real analysis and encountered this problem. Prove that for almost everywhere $x\in\mathbb{R}$, $\lim_{n\rightarrow\infty}|\cos{nx}|^{\frac{1}{n}}=1$. What theorem can I use to solve this problem? I don't know how to start. Thanks.
user709182
7
votes
1 answer

Show that $\frac{f(x)}{x}$ is a decreasing function implies that $f(x)$ is subadditive

I am studying Carother's Real Analysis for my qualifying exams. In the book I am to prove that if $f : [0, \infty) \rightarrow [0, \infty)$ is an increasing function, $f(0) = 0$, and $f(x) > 0$ for all $x > 0$, then $\frac{f(x)}{x}$ being decreasing…
Aphyd
  • 808
7
votes
3 answers

Prove that $\sum_{k=1} ^ n k^{-1} = \ln(n) + O(1)$

I would like to prove that $\sum_{k=1} ^ n k^{-1} = \ln(n) + O(1)$. That is, I would like to show that there is some natural number $N$ large enough so that $n \ge N$ implies: $$|\sum_{k=1}^n k^{-1} - \ln(n) | < M,$$ where $M$ is a positive constant…
JZS
  • 4,844
7
votes
1 answer

Difference between gradient and derivative.

My question may be a bit stupid, but this morning I tried to explain the gradient to someone, and he makes a parallel with derivative of function $f:\mathbb R\to \mathbb R$. What he says is that for a function $f:\mathbb R\to \mathbb R$, the…
user657324
  • 1,863
7
votes
2 answers

Compactness of finite sets

In rudin's analysis books, he defines compactness as: A subset $K$ of a metric space $X$ is ${\bf compact}$ if every open cover of $K$ contains a finite subcover. More explicitly is that if $\{ G_{\alpha} \}$ is an open cover of $K$ then one can…
ILoveMath
  • 10,694
7
votes
2 answers

A multi variable function that satisfies 3 conditions

Let $f(x,y)$ be multi variable function that is defined when $\frac{x}{2} f(x,y)$ 2 ) $f(x-1,y-\ln x) > f(x,y)(1-\frac{1}{x})$ 3 ) $|f(x,y) - \ln x| <…
Ahmad
  • 1,380
  • 1
  • 18
  • 37
7
votes
1 answer

Find $\displaystyle \min_{f \in \mathcal{A}} \int_{0}^{1} (1+x^{2})(f(x))^{2} dx $

Find $$ \min_{f \in \mathcal{A}} \int_{0}^{1} (1+x^{2})(f(x))^{2} dx $$ where $$ \mathcal{A}=\left\{ f \in C[a,b] \ | \ \int_{0}^{1} f(x) dx = 1 \right\}.$$ I used the Hölder's inequality to try to solve the problem but I do not know how to reduce…
Jesus Carreño
  • 149
7
votes
2 answers

Proving a specific set in $\mathbb R^2$ is closed.

I am trying to prove that the set $A=\{(x,y)|x^2\leq y\}$ is closed in $\mathbb R^2$. I wrote a proof, but I think the end is wrong. My proof is: Consider the set $A=\{(x,y)|x^2\leq y\}$ in $\mathbb R^2$. Let $(x_n,y_n)_{n \in \mathbb N}$ be a…
CleoFroPollo
  • 111
7
votes
2 answers

Fubini's theorem for Riemann integrals?

The integrals in Fubini's theorem are all Lebesgue integrals. I was wondering if there is a theorem with conclusions similar to Fubini's but only involving Riemann integrals? Thanks and regards!
Tim
  • 47,382
7
votes
3 answers

Is the Euclidean Norm Differentiable at $ 0 $?

As the title, is the euclidean norm differentiable at $0$? I tried that prove by contradiction and apply the definition but can't really get a contradiction. Any hints?
Johnny
  • 231
7
votes
4 answers

Proving that a set is complete

I am working on a question and I am looking for some clarification. I can't seem to use what I know to complete the proof. Let $\mathbf{x}_{0} \in \mathbb{R}^n$ and $R>0$. Prove that $U=\left \{ \mathbf{x} \in \mathbb{R}^n : \left \| \mathbf{x} -…
clarence_atwater
  • 153
7
votes
4 answers

Difference between $\epsilon-n_0$ and $\epsilon-\delta$ limit definition.

My Real Analysis course uses the $\epsilon - n_0$ definition of the limit, but I have noticed that the $\epsilon - \delta$ approach seems to be more common. Could someone please explain both formally and informally the difference between the…
economista
  • 155
1 2 3
…
99 100