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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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Prove existence of $c$ such that $f(x)\ge c g(x)$

Let $f$ and $g$ be non-negative continuous functions on $[0,1]$ such that $f(x)>g(x)$ for all $x$ in $[0,1]$. Show that there exists a constant $c>1$ such that for all $x$ in $[0,1]$ we have $f(x)\ge c g(x)$. I tried using the fact that since…
mikefallopian
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Fubini's theorem problem

Let $f$ be a non-negative measurable function in $\mathbb{R}$. Suppose that $$\iint_{\mathbb{R}^2}{f(4x)f(x-3y)\,dxdy}=2\,.$$ Calculate $\int_{-\infty}^{\infty}{f(x)\,dx}$. My first thought was to change the order of integration so that I…
Bey
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Sequence in uncountable set

Given an uncountable subset $A$ of $(0,1)$, does there always exist $a,r>0$ such that $a+r,a+r^2,a+r^3,\dots$ are all in $A$? For example, if $A$ contains an interval, it is easy to find such $a,r$. To try to show this, we can assume that no such…
user57012
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Help with $ \lim\limits_{t \to 0} \int_{-1}^1 \frac{t}{t^2+x^2} f(x)\ dx$

Show that if $f$ is continuous on $[-1,1]$, then $$ \lim_{t \to 0} \int_{-1}^{1}\frac{t}{t^2+x^2}f(x)\,dx=\pi f(0) $$ Any hints?
sl97
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If $f(x)\to +\infty$ as $x\to +\infty$, then why is $\lim_{x\to \infty}\frac{\sin{(x^2+x+1)}}{f(x)}=0$?

If $f(x)\to +\infty$ as $x\to +\infty$, then $$\frac{\sin{(x^2+x+1)}}{f(x)}\to 0, \qquad \text{ as } x\to+\infty$$ I know the following is true by the Squeeze Theorem. I am just not sure how to apply it. Any suggestions?
Liza
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Closed property of nonempty finite set

I came across this text in Rudin's book where it has been mentioned that a non-empty finite set is closed. But a closed set is a set which contains all of it's limit points in the set itself but none of the elements of a non-empty finite set can…
Aashish Loknath Panigrahi
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Prove that $\lim_{x \rightarrow 0} \frac{1}{x}\int_0^x f(t) dt = f(0)$.

Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. Prove that $\lim_{x \rightarrow 0} \frac{1}{x}\int_0^x f(t) dt = f(0)$. I'm having a little confusion about proving this. So far, it is clear that $f$ is continuous at 0 and $f$ is…
tk2
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Prove the unit circle is uncountable

This is a homework problem, so avoid giving the answer. I think a discussion of my attempt at a proof would be more appropriate. The problem goes as follows: Let $S$ be the circle of unit radius in the Euclidean plane: $$S = \{ (x,y) \in…
CrypticParadigm
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Find limits of sequences

Prove that $$a) \lim\limits_{n \to \infty} (\frac{1^p+2^p+...+n^p}{n^p} - \frac{n}{p+1})=\frac{1}{2},$$ $$b) \lim\limits_{n \to \infty} \frac{1^p+3^p+...+(2n-1)^p}{n^{p+1}}=\frac{2^p}{p+1},$$ where is $p \in \Bbb N $. Thanks to Stolz–Cesàro theorem…
theuses
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Linear combinations of sequences of uniformly integrable functions

Let $\{f_n\}$ and $\{g_n\}$ be sequences of uniformly integrable functions on $E$. Show that for $\alpha$, $\beta$, the sequences $\alpha f_n + \beta g_n$ are also uniformly integrable. Attempt at a proof: Since both sequences are uniformly…
emka
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if $f:[0,1] \to \mathbb{R}$ is increasing, show that $f$ is the pointwise limit of a sequence of continuous functions over $[0,1]$

if $f:[0,1] \to \mathbb{R}$ is increasing, show that $f$ is the pointwise limit of a sequence of continuous functions over $[0,1]$ Intuitively this makes sense but I am having trouble with showing why there would be a sequence of continuous…
oliverjones
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does a convex polynomial always reach its minimum value?

Consider a convex polynomial $p$ such that $p(x_1,~x_2,\dots x_n)\geq 0~\forall x_1,~x_2,\dots x_n\in \mathbb{R}^n$. Does the polynomial reach its minimum value? This is not true for non-convex polynomials like $(1-x_1x_2)^2+x_1^2$, see the response…
Daniel Porumbel
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Modification of Dini's theorem

Classical Dini's theorem states that if $(f_n)$ is a monotone sequence of continuous functions on a compact space converges pointwise to a continuous function $f$, then the convergence is uniform. It is shown that all conditions are needed for the…
Idonknow
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How can I prove that this sequence is monotonic?

I have a sequence $(u_n)$ that is defined as: $u_0 = 2$, $u_{n+1} =\frac{u_n}{2} + \frac{1}{u_n}$ I have tried to prove that it is monotonic using induction but I wasn't able to succeed. How can I prove it easily ? Thank you
Pop Flamingo
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Rudin Series ratio and root test.

In Rudins Principles of Mathematical Analysis he says consider the following series $$\frac 12 + \frac 13 + \frac 1{2^2} + \frac 1{3^2} + \frac 1{2^3} + \frac 1{3^3} + \frac 1{2^4} + \frac 1{3^4} + \cdots$$ for which $$\liminf \limits_{n \to…
Differintegral
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