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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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Suppose $(a_n)$ is a sequence such that $a_n=\frac{1!+2!+\cdots+n!}{n!}$. Show that $\lim{a_n}=1$

Suppose $(a_n)$ is a sequence such that $$a_n=\frac{1!+2!+\cdots+n!}{n!} \, .$$ Show that $\lim_{n \rightarrow \infty}{a_n}=1$. My attempt is to formulate an inequality and then use the Squeeze Theorem. Since we know that…
Idonknow
  • 15,643
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can polynomial functions with rational coefficients approximate any continuous function on $[a,b]$.

The Stone-Weierstrass theorem states that for any continuous function $f$ on $[a,b]$ there exists a polynomial function $p$ such that $\|f-p\|<\varepsilon$ for any $\varepsilon>0$. Where $\|\cdot\|$ is the sup norm. If I replace "a polynomial…
user1292919
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How to prove that $\lim(\underset{k\rightarrow\infty}{\lim}(\cos(|n!\pi x|)^{2k}))=\chi_\mathbb{Q}$

Possible Duplicate: How is this called? Rationals and irrationals Please help me prove, that $$\underset{n\rightarrow\infty}{\lim}\left(\underset{k\rightarrow\infty}{\lim}(\cos(|n!\pi x|)^{2k})\right)=\begin{cases} 1 & \iff x\in\mathbb{Q}\\ 0 &…
Steve
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Is the set of all polynomials in $\log(x)$ dense in $L^2[0,1]$?

Is $\{(\log(x))^k\mid k=0,1,2,\ldots\}$ dense in $L^2 [0,1]$? That is, is the set of all polynomials of logarithm functions dense in the set of square integrable functions on $[0,1]$?
LWW
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Proof of Lemma 8.5.14 in Terence Tao Analysis I

Lemma 8.5.14. Let X be a partially ordered set with ordering relation $\leq$, and let $x_0$ be an element of $X$. Then there is a well-ordered subset $Y$ of $X$ which has $x_0$ as its minimal element, and which has no strict upper bound. Proof. The…
bin
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Continuous function and Hausdorff space

The question is: $f: X \to Y$ is continuous bijection, which of the following is correct: I. if $X$ is Hausdorff space then $Y$ is Hausdorff space. II. if $X$ is compact and $Y$ is Hausdorff space, then $f^{-1}$ exist. I think I is correct since…
astr627
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Everywhere Super Dense Subset of $\mathbb{R}$

First, this question is motivated by the imprecise question: Is there a sensible notion of parity (evenness and oddness) for real numbers? Here are some properties a notion of parity for $\mathbb{R}$ should have: It should be an equivalence…
Joshua Ruiter
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how to prove, $f$ is onto if $f$ is continuous and satisfying $|f(x) - f(y)| ≥ |x - y|$ for all $x,y$ in $\mathbb{R}$

let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that $|f(x) - f(y)| ≥ |x - y|$ for all $x,y$ in $\mathbb{R}$. then $f$ is onto.
ram
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Uniform semi-continuity

Background It is a standard and important fact in basic calculus/real analysis that a continuous function on a compact metric space is in fact uniformly continuous. That is, suppose $(X,d)$ is a compact metric space and $f\colon X \to\mathbb R$ is…
Vaughn Climenhaga
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Contiuous function on a closed bounded interval is uniformly continuous. Don't understand the proof.

I'm self studying real analysis from Wade's "An Introduction to Real Analysis" and I've come across a proof that I don't understand. I was hoping that some might be able to walk me through it. The theorem is as follows Theorem. Suppose that $I$ is a…
mark
  • 1,751
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Proving continuity and monotonicity of $t\mapsto t^x, t>0$ with minimal assumptions.

I'm trying to prove that The function $t\mapsto t^x,\, x\in \Bbb R,\, t>0$ is continuous and monotonic. Suppose $+, \cdot\,:\Bbb R^2\to \Bbb R$ (addition and multiplication) have already been defined (via the standard dedekind cuts…
YoTengoUnLCD
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Characterization of real functions which have limit at each point

The following problem is Exercise 7.K from the book van Rooij-Schikhof: A Second Course on Real Functions and it is very close to a question which was recently discussed in chat. So I thought that sharing this interesting problem with other MSE…
Martin Sleziak
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Is Koch snowflake a continuous curve?

For Koch snowflake, does there exits a continuous map from $[0,1]$ to it? The actural construction of the map may be impossible, but how to claim the existence of such a continuous map? Or can we conside the limit of a sequence of continuous map,…
Sun
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Continuous functions are bounded

How one demonstrates this claim: Let $F:\mathbb{R}^m \rightarrow \mathbb{R}^n$ be a continuous function, if $X \subset \mathbb{R}^m$ is bounded then $F(X)$ is bounded.
Jr.
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Proving that Continuous Open Functions are Strictly Monotonic

It is a fact from analysis that a continuous and open real-valued function of a real variable is strictly monotonic. The proof I know runs something like this: Suppose $f$ is an open and continuous map but is not strictly monotonic. Consequently,…
ItsNotObvious
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