# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

93,829 questions
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### Prove that $(z_n)_n$ is bounded $\Rightarrow$ $(w_nz_n)_{n\geq 1}$ is a null sequence (a sequence tending to $0$)

Let $z\in \mathbb{C}$, $(z_n)_{n\geq 1} \subset \mathbb{C}$ and $(w_n)_{n\geq 1}$(a null sequence) be sequences. Prove that $(z_n)_n$ is bounded $\Rightarrow$ $(w_nz_n)_{n\geq 1}$ is a null ...
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### Calculate $\lim _{x \rightarrow 0} \frac{\int _{0}^{\sin x} \sqrt{\tan x} dx}{\int _{0}^{\tan x} \sqrt{\sin x} dx}$

Calculate $$\lim _{x \to 0}\frac{\int_{0}^{\sin\left(x\right)} \,\sqrt{\,\tan\left(t\right)\,}\,\mathrm{d}t} {\int_{0}^{\tan\left(x\right)}\,\sqrt{\,\sin\left(t\right)\,}\, \mathrm{d}t}$$ I have ...
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### Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ is a function. And $f_{xx}$ and $f_{yy}$ exist and continuous. Is $f$ a twice differenced function?

Obviously, it is true if $\delta>0$ imply $2f(x+\delta, y+\delta)-f(x+2\delta, y)-f(x, y+2\delta)=o(\delta^2)$. But this result I can't prove too.
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### Characterization of measurable functions mapping into banach space via simple functions?

This is a statement I read in Lang's Real and Functional Analysis Chpt VI, Sec 1, M8. M8. A map $f:X\to E$ with $E$ finite dimensional real vector space is measurable if and only if $f$ is a limit ...
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### Is there an exception to the law of large numbers?

I was reading about the law of large numbers, and under its strong formulation it says that the sample average converges almost surely. That means that it may exist a finite subset with measure $0$ ...
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### Rank of differential (Jacobian) is local minima in continuously differentiable functions

Let $f\in C^1(\mathbb{R}^n\to\mathbb{R}^m)$ and let $a \in \mathbb{R}^n$ Prove there exists an open ball $B(a,\epsilon)$ such that for every $x\in B(a,\epsilon)$: $rankD_f(a)\le rankD_f(x)$ ...
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### Solving an infinite sum using integration or derivation

There is an infinite sum given: $$\sum_{n=1}^{\infty}\frac{1}{n^22^n}$$ It should be solved using integration, derivation or both. I think using power series can help but I don't know how to finish ...
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### Find $\lim_{(x, y)\to(1, 1)}\frac{x^y-y(x-1)-1}{(x-1)^2 + (y-1)^2}$

$\lim_{(x, y)\to(1, 1)}\frac{x^y-y(x-1)-1}{(x-1)^2 + (y-1)^2}$ I am trying to use squeeze theorem but I am having trouble with finding an upper bounding convergent to 0 (I believe that 0 is the limit)...
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### Change the order of minimum in sums

Is it possible to change the order of minimum as i did below: $$\min_w \sum_x\sum_y f(x,y,w)^2 = \sum_x \min_w \sum_y f(x,y,w)^2$$ If for all $x$ we have $$\min_w \sum_y f(x,y,w)^2,$$ then could I ...
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### Odd Bernoulli numbers are zero using functional equation of Bernoulli Polynomials

Let $(B_k)$ denote the sequence of Bernoulli numbers and let $B_n(x):=\sum_{k=0}^n {n\choose k}B_k x^{n-k}$ denote the Bernoulli polynomials. Bernoulli Polynomials satisfy the following functional ...
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### How to evaluate $\lim _{x\to \infty }\left(1+2x\sqrt{x}\right)^{2/\ln x}$?

I have a problem with this limit, can you please show me a way to solve it without L'Hôpital's rule? $$\lim _{x\to \infty}\left(1+2x\sqrt{x}\right)^{\frac{2}{\ln x}}$$ This is my solution (it's ...
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### $\lim_{x \to 0} \frac{e^{2x} - \ln(1-x) - \sin(x)}{\cos(x)-1}$ using Taylor Expansions

As a preface- a very similar question is here: Using Taylor expansion to find $\lim_{x \rightarrow 0} \frac{\exp(2x)-\ln(1-x)-\sin(x)}{\cos(x)-1}$ But, my actual question is, when we substitute the ...