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
8
votes
4 answers

Is The Union of Intervals an Interval or not?

I have a question on union of intervals. My teacher says the union of intervals is not an interval. Is this always true? I mean $[0,2] \cup [4,5]$ is not an interval, because $[0,2] \cap [4,5]= \varnothing ,$ but what about $ (0,8) \cup (7,9).$ I…
Dima
  • 2,479
8
votes
2 answers

Given subsequences converge, prove that the sequence converges.

I have looked through previous posts but have been struggling with this problem. The sequence is {$a_n$} and its subsequences {$a_{2k}$}, {$a_{2k+1}$}, {$a_{3k}$} converge. I have to prove that {$a_n$} converges. I know that a sequence converges if…
ozarka
  • 499
8
votes
3 answers

Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.

Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$. So $(a^2+b^2)(c^2+d^2) = a^2c^2+a^2d^2+b^2c^2+b^2d^2$ and $(ac+bd)^2 = a^2c^2+2acbd+b^2d^2$ So the problem is reduced to proving that $a^2d^2+b^2c^2\ge2acbd$ but I am not sure…
Burgundy
  • 2,097
8
votes
4 answers

Can you take Dedekind Cuts of the real numbers?

My professor defined Dedekind cuts in the following way: "Given two nonempty sets E,F $\subset \mathbb{R}$, we say that the pair (E,F) is a Dedekind cut of $\mathbb{R}$ if $E \cap F = \emptyset $ $E \cup F = \mathbb{R}$ $x < y$ for all $x \in…
Physics_Plasma
  • 267
8
votes
2 answers

In $C[0,1]$ prove that the subset of Lipschitz functions is dense

In $C[0,1]$ prove that the subset of Lipschitz functions is dense. I can't prove it.
Gaston Burrull
  • 5,449
  • 5
  • 33
  • 78
8
votes
2 answers

Showing $\sin^n(x)$ does not converge uniformly on $[0,\pi/2]$

I just want to check that I am correct in my argument that $f_n(x) = \sin^n(x)$ does not converge uniformly. When $x = \pi/2$, $\sin^n(x) = 1$ for all $n$, hence $ f_n(\pi/2) \rightarrow 1$, However, for all other $x$, $f_n(x) \rightarrow 0$. Hence…
fosho
  • 6,334
8
votes
2 answers

How do I prove that $f_n\to f$ in $L^p$?

Let $\{f_n\}$ be a sequence in $L^p([0,1])$ for $p\geq 1$. Suppose there exists $f\in L^p([0,1])$ satisfying $\lim_{n\to\infty} \int_0^1 f_n(x)g(x)dx = \int_0^1 f(x)g(x)dx$ for any $g\in L^2([0,1])$. Moreover, assume that $\lim_{n\to\infty}…
Rubertos
  • 12,491
8
votes
2 answers

Changing the order of $\lim$ and $\sup$

Suppose that $f_n:X\to [0,1]$ where $X$ is some arbitrary set. Suppose that $$ f_n(x)\geq f_{n+1}(x) $$ for all $x\in X$ and all $n = 0,1,2,\dots$ so there exists $\lim_n f_n(x)$ point-wise, let's call it $f(x)$. Define $f^*_n:=\sup\limits_{x\in…
SBF
  • 36,041
8
votes
3 answers

Does $|f'(x)|<1$ imply $f$ has a fixed point?

$f :\mathbb R \rightarrow \mathbb R$ is differentiable on $\mathbb R $ and $|f'(x)| \lt 1$, does $f$ have a fixed point? I think it does but I can't finish the proof. Let's define $g(x) = f(x) - x$, we want to prove that this function is equal to…
Bag of Chips
  • 325
8
votes
1 answer

Show that the set of points where a real-valued continuous sequence of functions converges is $F_{\sigma\delta}$

By $F_{\sigma\delta}$, I mean that the set can be expressed as a countable intersection of $F_\sigma$ sets. Let this sequence of functions be $f_n$, and the set of points where $f_n$ converges be $C$. Since $f_n$ must be Cauchy, I can define $C$…
Jake Browning
  • 401
8
votes
2 answers

countable or uncountable sets NBHM 2016

The set of all algebraic numbers the set of all strictly increasing infinite sequences of positive integers the set of all infinite sequences of integers which are in arithmetic progression . I know 1 is definitely countable set. and i am not…
himanshu Singla
  • 155
8
votes
2 answers

Show that the square root of a non-negative operator is unique

Let $H$ be a Hilbert space, and $A\in B(H\to H)$ be a bounded non-negative operator (i.e. $\langle Ax,x\rangle \geq 0$ for all $x\in H$). The square root of $A$ is a bounded non-negative operator $B\geq $ such that $B^2=A$. First, We can assume…
Xiang Yu
  • 4,835
8
votes
2 answers

How do I prove a sequence is Cauchy

I was hoping someone could explain to me how to prove a sequence is Cauchy. I've been given two definitions of a Cauchy sequence: $\forall \epsilon > 0, \exists N \in \mathbb{N}$ such that $n,m> N$ $\Rightarrow |a_n - a_m| ≤ \epsilon$ and…
CrossGuard
  • 297
8
votes
2 answers

What am I doing wrong in this proof?

The question is this: Let $f:\mathbb{R}\to\mathbb{R}$ be differentiable at $x=0$ and suppose that there is a number $L$ such that $$\lim_{x\rightarrow0}\frac{f(x)-f(x/2)}{x/2}=L.$$ Prove that $f'(0)=L$. Here's my answer with all theorems referenced…
Andrew V
  • 83
8
votes
1 answer

If the limit of a derivative is zero as $x \to \infty$, what can we say about $f(x+1)-f(x)$?

Given a differentiable function $f$ such that $$ \lim_{x \to +\infty}f'(x) = 0 $$ what can we say about $$ \lim_{x\to\infty}(f(x+1)-f(x)) \text{ ?} $$ My first thought was to use mean value theorem on $[x,x+1]$, and I will get that the limit is…
Butterfly
  • 1,443
1 2 3
…
99 100