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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. .

145439 questions
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Does one of $L^\infty$ and $L^p, p \in (0, \infty)$ contain the other?

I think $L^\infty(\Omega, \mathcal{F}, \mu)\supseteq L^p(\Omega, \mathcal{F}, \mu), \forall p \in (0, \infty)$? My reason is $L^\infty$ is defined as the set of measurable functions that are bounded up to a set of measure zero, and if $f \notin…
Tim
  • 47,382
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Helly's selection theorem (For sequence of monotonic functions)

Let $\{f_n\}$ be a sequence of monotonically increasing functions on $\mathbb{R}$. Let $\{f_n\}$ be uniformly bounded on $\mathbb{R}$. Then, there exists a subsequence $\{f_{n_k}\}$ pointwise convergent to some $f$. Now, assume $f$ is continuous on…
Katlus
  • 6,593
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Prove a set is at most countable

Suppose that for each $\lambda$ in a set $\Lambda$ we have a positive real number $a_\lambda > 0$. Suppose also that for any natural number $n$ and any $\lambda_1, \cdots, \lambda_n \in \Lambda$ we have $$ \sum_{ i = 1 }^n a_{ \lambda_i } < 1…
Jones Monez
  • 143
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8 answers

How to define the $0^0$?

Possible Duplicate: Zero to zero power According to Wolfram Alpha: $0^0$ is indeterminate. According to google: $0^0=1$ According to my calculator: $0^0$ is undefined Is there consensus regarding $0^0$? And what makes $0^0$ so problematic?
Kasper
  • 13,528
6
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4 answers

Prove that $\sum_{k=1}^{n} \frac{1}{n+k} < \ln 2$

I have a loose upper bound: $$ \sum_{k=1}^{n} \frac{1}{n+k} < \frac{1}{n} \cdot n = 1 $$ Clearly this is very loose compared to the upper bound $\ln 2 \approx 0.693$
seeker
  • 297
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2 answers

is it possible to find the closest rational number to an irrational number?

given an irrational number is it possible to find the closest rational number to the irrational number? If so, how?
user1066113
  • 335
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Let $f(x)$ be differentiable over $\left[0,1\right]$ with $f(0) = f(1) = 0$. Prove that $f'(x) - 2f(x)$ has a zero in $(0,1)$

for a final exam review for Real Analysis, the following problem is asked: Let $f(x)$ be a differentiable function over $[0,1]$. Suppose that $f(0) = f(1) = 0$. Prove that $f'(x) - 2f(x)$ must have a zero inside of $(0,1)$. My initial idea was to…
EnCt272cf2928267cbbc
  • 61
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Arzela Ascoli counterexamples

I am looking for some examples that show that the Arzela Ascoli theorem is "tight". i.e. is there a sequence of functions that is uniformly bounded and equicontinuous on a noncompact set that would not have a uniformly convergent subsequence. Also…
GTOgod
  • 570
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If limit of $|f|$ is divergent to $\infty$, then is $f$ divergent to either $\infty$ or $-\infty$?

Let $f$ be a real function and $a$ be a limit point of its domain. Suppose $\lim_{x\to a} |f(x)| = \infty$. How do i prove that if $f$ is continuous on its domain, its limit is either $\infty$ or $-\infty$? (And Its domain is connected) I think…
Katlus
  • 6,593
6
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1 answer

If $f(\mathbb{R}) \subseteq \mathbb{Q}$, prove that $f$ is constant.

Let $f:\mathbb{R}\to \mathbb{R}$ is a function with $f(\mathbb{R})\subseteq \mathbb{Q}$ such that for every Cauchy sequence of rational numbers $(a_i)$, $\lim_{i\to \infty}f(a_i)$ exists. Prove that $f$ is constant. If I can prove that $f$ is…
XYZABC
  • 1,053
6
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2 answers

An application of the Mean Value Theorem

I'm recalling this question from memory, so I may be messing it up a bit. Let $a/3+b/2+c=0$. Show that $ax^2+bx+c=0$ has at least one root in $[0,1]$ using the Mean Value Theorem. Let $f(x)=ax^2+bc+c$. Then $f(0)=c$ and $f(1)=a+b+c$. Also…
emka
  • 6,494
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3 answers

For every real $x>0$ and every integer $n>0$, there is one and only one real $y>0$ such that $y^n=x$

I am unable to prove that For every real $x>0$ and every integer $n>0$, there is one and only one real $y>0$ such that $y^n=x$. Can anyone please help me here? It is clear that there is at most one such real $y$. But how do I go about the existence…
Richard Nash
  • 639
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Definition of $\limsup$ and $\liminf$ for functions

In my notes, I have written that \begin{align} \limsup_{x\to a} f(x) & = \inf_{\delta\to0}\sup\left[f(x): \Vert x-a \Vert < \delta \right] \\ & = \inf_{n\to \infty}\sup\left[ f(x): \Vert x-a \Vert <\frac{1}{n} \right] \\ & = \lim_{n\to…
Lemon
  • 12,664
6
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1 answer

Series unchanged by rearrangement implies absolute convergence?

If a series converges absolutely, then it is known that the value of the series is independent of rearrangements. More precisely, if $\sum |a_n| < \infty$ and $\sigma:\Bbb N\to \Bbb N$ is a bijection then $\sum a_{\sigma(n)}$ converges and its value…
Alex Ortiz
  • 24,844
6
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4 answers

Prove $f$ is constant if $|f(x)-f(y)|\leq M|x-y|^\alpha$

Let $\alpha > 1$ and $M \geq 0$. Suppose $f: \mathbb{R} \longrightarrow \mathbb{R}$ satisfies $|f(x)-f(y)|\leq M|x-y|^\alpha$ for all $x, y\in \mathbb{R}$. How can we prove that $f$ is a constant function? I don't even know where to start.
Joakim
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