# 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
1answer
62 views

1answer
44 views

1answer
25 views

0answers
27 views

### 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 ...
0answers
23 views

### 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 ...
1answer
42 views

### 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 $(0,1)$...
1answer
22 views

### subset of a special type of a set [on hold]

Any help is appreciated. Can't process the hint.
3answers
50 views

### 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.
0answers
29 views

### 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)...
1answer
43 views

### 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 ...
0answers
72 views

0answers
30 views

### 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 ...
2answers
90 views

### 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 : $\sum_{k=1}^{n}k^{2}=\frac{n(n+1)(2n+1)}{6}$
3answers
27 views

### 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 ...
1answer
49 views

### 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 ...
0answers
55 views

2answers
43 views

### 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 $A$.
1answer
47 views

### Definition of the infimum of a set in terms of a minimum.

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 ...
1answer
28 views

### Riemann Stieltjes Mean Value theorem - result

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 ...