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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 $f$ is differentiable on $(0, \infty)$ and $\lim_{x \to \infty} f'(x) = 0$

Suppose $f$ is differentiable on $(0, \infty)$ and $\lim_{x \to \infty} f'(x) = 0$. I want to prove that $f$ is uniformly continuous on $[1, \infty)$. So suppose the hypothesis so we have $\lim_{x \to \infty} f'(x) = 0$. Now there must exist $M$…
user247618
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Compact sets in $\mathbb R$ are closed

What is the best way to prove the statement? I know the version of using nested intervals, but is there another way to approach the problem?
cooselunt
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Problem 19 chapter 4 from baby Rudin

Suppose $f$ is a real function with domain $\mathbb{R}^1$ which has the intermediate value property: If $f(a)
RFZ
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Arbitrary intersection of compact sets is compact.

To prove this, I want to show that an arbitrary intersection of closed sets is closed and an arbitrary intersection of bounded sets is bounded. I know how to prove the first part, but I'm not sure how to rigorously show that an intersection of…
Moz
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Algebraic Proof of Geometric Claim

I derived this claim geometrically: If $a
wjmolina
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Iterated integrals agree and finite but double is infinite

What would be an example of $f$ s.t. $\int_0^1 \int_0^1 f dx dy = \int_0^1 \int_0^1 f dx dy < \infty$ but the double integral $$\int_{[0,1]^2} f \ d(x,y) = \infty$$ Or maybe there doesn't exist such $f$ (which I doubt)? The integrals are Lebesgue.
user16015
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Mean value theorem and second derivative

right now I'm reading a paper by Blower (Displacement convexity for generalized orthogonal ensemble) and I'm stuck at a rather delicate point. He uses the following equality, which is supposed to be deducable from the mean value theorem: Lemma: Let…
Arthur Sinulis
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given A is open and B subsets.to show A+B open

Q. i.prove that if A is open and B is arbitrary subset of $R^n$ then A+B ={x+y : $x\in A$, y $\in B$ } is open. ii.show that if A and B are closed subsets of R then A+B need not be closed. my doubt: in this question do i have to show that there…
Foggy
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Find local extrema of the following function.

Find local extrema of the function $$u(x,y,z)=\sin x \cdot \sin y\cdot \sin z$$ with the condition $$x+y+z=\frac{\pi}{2};\; x,y,z>0$$ Can anyone give me pointers on how to solve this problem?
zermelovac
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How Euler get $\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$.

I saw on wikipedia that he consider $$\frac{\sin(x)}{x}=1-\frac{x^2}{3!}+\frac{x^4}{5!}+...$$ The roots are given by $x=\pm n\pi$ and thus (to me) $$\frac{\sin x}{x}=(x-\pi)(x+\pi)(x-2\pi)(x+2\pi)(x-3\pi)(x+3\pi)...$$ How can he get $$\frac{\sin…
idm
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Does Darboux property imply strong Darboux property?

Given a set $S$, we say that a monotone set function $f\colon \mathcal{P}(S) \to [0,1]$ has the strong Darboux property whenever $f(\emptyset)=0$, $f(S)=1$, and for all $A\subseteq B \subseteq S$ and $x \in [f(A),f(B)]$ there exists $A\subseteq…
Paolo Leonetti
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Separate continuity does not imply joint continuity, but where am I going wrong?

Given a function $f(x,y)$, separate continuity means fixing $y=y_0$ then $f(x,y_0)$ is continuous with respect with $x$, and fixing $x=x_0$ then $f(x_0,y)$ is continuous with respect with $y$. Joint continuity simply means $f(x,y)$ is continuous. It…
Tony
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Computing $\sum_{n=-\infty}^\infty \int_{-\pi}^\pi e^{-x^2} \cos(nx) dx $

I'm trying to find $$ \sum_{n=-\infty}^\infty \int_{-\pi}^\pi e^{-x^2} \cos(nx) dx. $$ About the only thing I can think of is the well-known identity $$ \sum_{n=-k}^k \cos(nx) = \sum_{n=-k}^k e^{inx} =…
Philip Hoskins
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Show that there is a metric space that has a limit point, and each open disk in it is closed. Collecting examples

Show that there is a metric space that has a limit point, and each open disk in it is closed. This question belongs to the 39th math competitions of Iran. This is one solution: Suppose that $X=\{\frac{1}{n}: n\in \mathbb{N}\} \cup…
lino
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Computing $\lim_{n \rightarrow \infty} n^2 \int_0^{2n} e^{-n \vert x-n \vert} \log \left[ 1+ \frac{1}{x+1} \right]dx$

I want to show that $$\lim_{n \rightarrow \infty} n^2 \int_0^{2n} e^{-n \vert x-n \vert} \log \left[ 1+ \frac{1}{x+1} \right]dx=2.$$ The reason I think this is the limit is because if we make the substitution $y = n(x-n)$, then we obtain $$…
Philip Hoskins
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