# 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.

93,656 questions
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### Showing that if $f = 0$ whenever $f=0$ on $\Omega \subset X$ then $\Omega$ is dense in $X$

I'm trying to solve this exercise in the lecture notes for a course in analysis that I'm taking. Proving that that defines a seminorm is straightforward. I'm only stuck in the step where I have to ...
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### Comparing two curves

For my Bachelor thesis, I have to analyse the goodness of an Approximation. The exact formula is called $\mathcal{I}(\varepsilon)$ and the Approximation $\mathcal{I}_{approx}(\varepsilon)$. In order ...
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### Confusion about a calculation concerning weak and strong convergence

I'm reading something and the consider the following example: Define: $$\chi(x) = \mathbb{1}_{[0,1/2]}-\mathbb{1}_{[1/2,1]}$$ and extend periodically to $\mathbb{R}$. Let $u_j = \chi(jx)$ on $(0,1)$...
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### subset of a special type of a set [on hold]

Any help is appreciated. Can't process the hint.
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### Limit of a sequence as $n\to\infty$ [on hold]

Let $x_{0}$ be a positive real number and $n\in\mathbb{N}$. Then what is $$\lim_{n\to\infty}\{(x_0+n)^r-n^r\}$$ where $r\in (0,1)$ is fixed number.
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### Difficult problems between triangle and square [on hold]

Problems : ABCD is a square centered at O. Let M be a point on the line segment [BD], and P and Q be the perpindicular projections of it on [AB], and [AD] respectively Prove that the lines (CM)...
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### Proving the divergence of $\sum_2^\infty{\frac{1}{n \log n}}$ using comparison test

It is quite straightforward using the Cauchy Condensation test. But is there any way to solve this problem using some well known comparison test? I cannot think of any way of my own. Any help/hint ...
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### When can we change the order of improper integral? [on hold]

Let $f(x,y)$ be continuous on $Q=\{(x,y)\mid x>0,y>0\}$ and $\iint_Q |f(x,y)|$ converge. Suppose $$\int_{0}^{\infty} f(x,y)\, \mathrm{d}y$$ converges uniformly on any finite interval. Show that ...
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### What is the smallest $n$ such that $\frac{n(n+1)(2n+1)}{6}$ is a square number? [on hold]

Question : Find the smallest natural number $n>1$ such that $\sum_{k=1}^{n}k^2$ is a square number Recall that : $\sum_{k=1}^{n}k^{2}=\frac{n(n+1)(2n+1)}{6}$
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### Root multiplicity for non-polynomial function

I know that if $f$ is a polynomial and $f(a)=0,f'(a)=0...,f^{(k)}(a)\neq 0$, then $a$ is a root of multiplicity $k$. Does this work for a differentiable function that is not a polynomial? I have seen ...
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### How would I start this problem?

Let $f \in C^2(\mathbb{R})$, let $v\in\mathbb{R}^n$ and $c>0$. Define a function $$u : \mathbb{R}^n\times \mathbb{R} \to \mathbb{R}, \qquad u(x,t) = f(v\cdot x - c\|v\|t)$$ and show that ...
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### Partial fraction expansion of $\frac{1}{(s+1)^{2}(s-1)(s+5)}$

I'm seeking a partial-fraction expansion of $A=\frac{1}{(s+1)^{2}(s-1)(s+5)}.$ I was solve equation differential using Laplace transform, but I need use partial fraction of $A$.
I have the following intuition regarding the relationship between the infimum of a set: Let's define the set of accumulation points of a set called $S$ as $S'$, its closure:$\bar{S}=S \cup S'$, and ...
This is a well known result of Riemann Stieltjes integration: All the proofs I found use the fact that $f$ is bounded and apply one the Mean Value Theorem Riemann-Stieltjes Integrals (this one). I ...