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
6
votes
2 answers

Show that $A \subset B \implies \overline{A} \subset \overline{B}$

NOTE: I know that a question asking for help to prove this same property already exists, but I would like an answer specifically based on the definition(s) and / or remark below, please. Definition 1: A point $x \in \mathbb{R}$ is a point of…
Isaac Bullock
  • 115
6
votes
1 answer

Representation of elements in Cantor set.

Every number in $[0,1]$ has a ternary expansion $$x=\sum_{k=1}^{\infty}a_k3^{-k}$$ where $a_k=0,1,$ or $2$.Note that this decomposition is not unique since, for example, $1/3=\sum_{k=2}^{\infty}2/3^k$.Prove that $x\in \text{Cantor set}$ if and only…
Laura
  • 4,689
6
votes
0 answers

Continuous functions on [0,1] with f(0) = f(1)

Possible Duplicate: Horizontal chord of length $\frac{1}{2}$ in the graph of a continuous function. Consider the set $C$ of real continuous functions defined on $[0,1]$ such that $f(0) = f(1)$. For each $f \in C$, we may consider the set $A(f) =…
Rick
  • 61
6
votes
1 answer

Why does $S = ([0,1] \times [0,1]) \cap \mathbb{Q}^2$ have no area?

I am currently in an Advanced Calculus 2 class and am using the C. H. Edwards "Advanced Calculus of Several Variables" text. In chapter 4 when discussing area in $\mathbb{R}^2$, the text says that the set $S = ([0,1] \times [0,1]) \cap…
realribs
  • 71
6
votes
2 answers

Does pointwise convergence against a continuous function imply uniform convergence?

Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions with $f_n:\mathbb{R}\rightarrow[0,1]$ for all $n\in\mathbb{N}$ and $f:\mathbb{R}\rightarrow[0,1]$ continuous such that $\lim_{n\rightarrow\infty}f_n(x)=f(x)$ for all $x\in\mathbb{R}$. Is it now…
stroem
  • 767
6
votes
0 answers

Constructive proof of a problem from the book Analysis by Terence Tao

Here is a problem from the book Analysis by Terence Tao: (Vol 1, Exercise 5.5.2) Let $E$ be a non-empty subset of $R$, let $n \geq 1$ be an integer, and let $L < K$ be integers. Suppose that $K/n$ is an upper bound for $E$, but that $L/n$ is not an…
Not an ID
  • 877
6
votes
4 answers

Open Cover / Real Analysis

I have the next question: Let $K \subset $ $R^1$ consist of $0$ and the numbers 1/$n$, for $n=1,2,3,\ldots$ Prove that $K$ is compact directly from the definition (without using Heine-Borel). I'm trying to understand compact sets so I would be…
Salieri
  • 283
  • 1
  • 14
6
votes
1 answer

space of riemann integrable functions not complete

Define norm as $\int |f|$ (Riemann integral) on $\mathcal R^1[0,1]$, the space of riemann integrable functions on $[0,1]$ with identification $f=g$ iff $\int |f-g|=0$. Let $\{ r_1,r_2,\cdots \}$ be the rationals in $[0,1]$, and let…
Gobi
  • 7,458
6
votes
1 answer

Proving integrability in integration by parts in Rudin's text

Integration by parts, as stated in W. Rudin's Principles of Mathematical Analysis, Theorem 6.22, goes as follows: Suppose F and G are differentiable functions in $[a,b]$, $F'=f\in \mathcal{R}$, and $G'= g\in \mathcal{R}$. Then $\int_a^bF(x)g(x)dx =…
mww
  • 179
6
votes
1 answer

Closing the loophole in this real analysis paper

In this paper, Theorem 1 states, given $F$ an arbitrary ordered subfield of $\mathbb{R}$, $F$ is complete iff every continuous function defined on a closed and bounded interval has a uniformly differentiable anti-derivative. The authors mention that…
user107952
  • 20,508
6
votes
3 answers

If $\lim\limits_{x \to \infty} f'(x) = L$ and $\lim\limits_{n \to \infty} f(n) = A$ exists, prove that $L = 0$.

Here is the homework problem I am stuck on: Let $f$ be differentiable on $(0,\infty)$. If $\lim\limits_{x \to \infty} f'(x) = L$ exists in $\mathbb{R}$ and $\lim\limits_{n \to \infty} f(n) = A$ exists in $\mathbb{R}$, prove that $L = 0$. From the…
FreshDresch
  • 160
6
votes
2 answers

What is the difference between total variation and arc length?

Let $f:[a,b] \rightarrow \mathbb R$ be $C^1$. Is the length of $\{f(x): x \in [a,b] \}$ and the total variation of $f$ the same thing ? The definition are extremely similar to each others: The total variation of a real-valued (or more generally…
W. Volante
  • 2,244
6
votes
1 answer

$2^x$ is irrational if $x$ is irrational?

Prove/Disprove that if $x$ is irrational, then $2^x$ is also irrational. My attempt for the proof: Suppose $2^x>0$ is a rational number, then $2^x=\frac{a}{b}$ for some natural numbers $a$ and $b$. Taking logarithm with base $2$ on both sides to…
Riaz
  • 2,174
6
votes
3 answers

How to show if $|a_{n+1} - a_{n}| \le \frac{1}{2^n}$ then the sequence is Cauchy.

Let $\{a_n\}$ be a sequence of real numbers such that $|a_{n+1} - a_n| \le \dfrac{1}{2^n}$. I would like to show that this sequence is Cauchy. Letting $\epsilon > 0$, I said choose $N$ such that $1/2^N \le \epsilon$. However, I'm not sure if…
user75120
  • 63
6
votes
4 answers

Is there an explicit solution to $a^x+b^x=1$?

Is there an explicit solution to $a^x+b^x=1$? Where $a, b \in [0, 1]$ and $a+b \le 1$. I've been playing around with this equation, but I can't seem to make any progress in solving it. I tried it in Wolfram for some hints, but nothing showed up.…
Stephen
  • 3,682
1 2 3
…
99 100