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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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An alternative proof to baby Rudin theorem 7.13

I am trying to prove Theorem 7.13 in baby Rudin and find that my proof goes in a totally different direction than the proof in the book. I briefly sketch what I did. Since $f_n(x)$ is continuous, then $f_n(x)$ is bounded in compact domain $K$…
kevin
  • 281
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$f(x)$ absolutely continuous $\implies e^{f(x)}$ absolutely continuous, for $x \in [a,b]$?

If $f(x)$ is absolutely continuous (a.c.) on [a,b], is the function $e^{f(x)}$ also absolutely continuous on [a,b] ? thanks
David
  • 183
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Prove that $\sum a_n$ converges iff $\sum 2^n a_{2^n}$ converges

Let $\{a_n\}$ be a positive, decreasing sequence. How do I prove that $\sum_{n=1}^\infty a_n$ converges iff $\sum_{n=1}^\infty 2^n a_{2^n}$ converges? I don't even know where to start. Any help would be appreciated.
user338193
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Is the axiomatic approach to defining $\mathbb{R}$ rigorous?

Some authors define $\mathbb{R}$ axiomatically. That is, they assume there exists a set $\mathbb{R}$ with binary operations $\cdot$ and $+$ such that the field axioms are satisfied, an order $\leq$ satisfying various axioms and the least upper bound…
MathematicsStudent1122
  • 12,462
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Is there a bijection between $[0, \infty)$ and $(0,1)$

I tried $\tan(x)$, and $\log(x)$, but seems it does not work, so I wonder is there a bijection or not?
qwerty
  • 71
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Direct proof of Bolzano-Weierstrass using Axiom of Completeness

The author of my intro analysis text has an exercise to give a proof of Bolzano-Weierstrass using axiom of completeness directly. Let $(a_n)$ be a bounded sequence, and define the set $$S=\{x \in \mathbb{R} : x
rorty
  • 1,181
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Refining Rudin's proof of $\lim \left (1+\frac 1 n\right)^n =\lim \sum_{k=1}^n \frac {1}{k!}$.

In PoMA Rudin writes Let $s_n=\sum_0^n \frac 1 {k!}$, $t_n=\left(1+\frac 1 n \right)^n$. By the binomial theorem, $$t_n=1+1+\frac{1}{2!}\left(1 - \frac{1}{n}\right) + \frac{1}{3!}\left(1 - \frac{1}{n}\right)\left(1 - \frac{2}{n}\right)…
YoTengoUnLCD
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Pointwise convergence implies $L^p$ convergence?

Let $f: X \to [0, \infty) \subset \mathbb R$ measurable where $X$ is a measure space. Let $f_n : X \to [0, \infty) $ be simple functions (i.e. linear combinations of characteristic functions of measurable sets) such that for each $x \in X$, $f_n(x)…
Rudy the Reindeer
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If a function has an antiderivative, is it always integrable?

If a function $f$ defined on $[a,b]$ has an antiderivative on $[a,b]$, is it always Riemann integrable? I can't think of a counterexample.
user315484
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"The deepest and most important of the fundamental properties of the real numbers" according to Edmund Landau

In his book Differential and Integral Calculus, Edmund Landau gives an introductory chapter, and before starting it, he assumes as true some theorems. Among them, it is one he calls: "The deepest and most important of the fundan1ental properties of…
Pedro
  • 122,002
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convolution is well-defined and differentiable for continuous $f$ and differentiable $g$ with compact support

Let $f$ be a continuous function from $\mathbb{R} \rightarrow \mathbb{R}$. Let $g \in C^1(\mathbb{R})$ with compact support. Prove that the convolution function $$(f*g)(x)=\int f(x+t)g(t) \,dt$$ is well-defined and lies in $C^1(\mathbb{R})$.…
user112358
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Little Rudin series convergence exercise

Problem: If $\sum a_n$ converges, and $\{b_n\}$ is monotonic and bounded, prove that $\sum a_n b_n$ converges. Source: Rudin, Principles of Mathematical Analysis, Chapter 3, Exercise 8.
Potato
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An application of intermediate value theorem to prove the existence of solutions

Let $f$ be a continuous function on $[0,1]$ with $f(0)=f(1)=0$. Prove that, for each $\alpha \in (0,1)$, there exists $x_1, x_2\in (0,1)$ such that $$f(x_1)=f(x_2)$$ with $x_1-x_2=\alpha$ or $x_1-x_2=1-\alpha$. The main difficulty in this problem is…
Richkent
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Prove that there isn't such a function, which would be continuous at each rational point and discontinuous in each irrational point.

Prove that there isn't such a function, which would be continuous at each rational point and discontinuous in each irrational point. I find this question very challenging and have no idea even how to start off with the proof. Please suggest a proof…
Stephen M.
  • 101
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Possible mistake in Rudin's definition of limits in the extended real number system

From Baby Rudin page 98 This seems to be a mistake since we have seemingly absurd results like $$ graph(f) = \{(0,0)\} \Rightarrow \lim_{x\to \infty} f(x)= 0 $$ We define the limit(for $x$ real) only for limit points of $E$ so my initial thinking…
B. Freitas
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