Proving a function $\rho:V\to[0,\infty)$ forms a norm on $V$ if and only if...
I have a function $\rho:V\to[0,\infty)$ where $V$ is a vector space and we know $\rho$ to be positive, absolutely homogeneous and non-degenerate (i.e.: we know $\rho$ satisfies all norm conditions...
View ArticleProve that there exists a point $e \in [0,3]$ with $f'(e) = 1/4$, given that...
I tried to prove this with Rolle's Theorem and with the Mean Value Theorem for Derivatives, yet no matter how I manipulated certain functions, I never arrived at that result. Any hint or help will be...
View ArticleProve that $\sum x^n$ does not uniformly converge for $x \in [0,1)$
I am trying to prove this by showing that $\lim_n \sup \{f - f_n : x \in [0,1)\} \neq 0$. I find that $f - f_n = \frac{x^n}{1-x}$. How can I show that the limsup of $\frac{x^n}{1-x}$ is not equal to 0?
View ArticleThe Carathéodory construction in comfortable instalments
I'm learning some (abstract) measure theory and consequently going through the bumpy road of $ 1/2^n $, $\pi $s and $ \lambda $s that starts from the notion of an outer measure on a set and ends with a...
View ArticleGood books to learn real analysis?? [duplicate]
I'm interested in learning real analysis. Do you have any book recommendations? I tried Rudin, but I found it challenging.
View ArticleGood book for self study of a First Course in Real Analysis
Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction...
View ArticleShow that the absolute value of an eigenfunction is again an eigenfunction
Let $\Omega \subseteq \mathbb{R}^n$ be open. Suppose $u : \Omega \to \mathbb{R}$ is a $C^2$ function satisfying $-\Delta u = \lambda u$ for some $\lambda \ge 0$. I would like to show that $|u|$ also...
View ArticleSuppose that $f$ satisfies $f(x+y)=f(x)+f(y)$, and that $f$ is continuous at...
Since $f$ is continuous at $0$...$$\forall \epsilon>0\exists \delta>0(\forall x\in \mathbb{R})[|x-0|<\delta\Longrightarrow |f(x)-0|]<\epsilon$$It is quite easy to see that $f(0)=0$ which...
View ArticleImage of the set where derivative vanishes has measure zero
Let, $f:[a,b]\to[c,d]$ be a strictly increasing function. Then, obviously it is differentiable almost everywhere. Now, consider the set $S=\{x\in [a,b]: f'(x)\text{ exists and }=0\}$, then does this...
View ArticleProving $\lim_{(x, y) \rightarrow (0, 0)} \frac{x^4 - y^4}{x^3 - y^3} = 0$...
Let $f(x, y) = \dfrac{x^4 - y^4}{x^3 - y^3}$ be defined in $U = \mathbb{R}^2 \setminus \{(x, x) \colon x \in \mathbb{R}\}$. I have to determine $\lim_{(x, y) \rightarrow (0, 0)} f(x, y)$ by...
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Estimate on derivative of a curve knowing diameter and second derivative
I'm reading the paper "A continuation principle for periodic solution of forced motion equations on manifolds and applications to bifurcation theory" by M. Furi and M. P. Pera.I'm not grasping the...
View ArticleProve that every well defined function satisfies $\epsilon-\delta$ definition...
The motivation for this question is a comment on this post, consider the logical formula:$$ ∀\epsilon \in \mathbb{R_{\geq 0}}, ∀x \in \mathbb{R}, ∃δ(x) \in \mathbb{R_{\geq 0}}:...
View ArticleProblem on showing a function is zero a.e.
Let $f\in L^1(\mathbb R)$ be such that$$\int_P f(y) dy =0$$for all Lebesgue measureable sets $P\subset \mathbb R$ with $m(P) = \pi$. Prove that $f(x)=0$ for almost all $x\in \mathbb R$.In attempt to do...
View ArticleStokes theorem on $C^1$ manifolds?
I have recently encountered Stokes theorem on embedded submanifolds of $\mathbb{R}^n$, and I didn't manage to find a proof for $C^1$ vector fields over $C^1$ manifolds, infact I have only seen that...
View ArticleProving that a function $f$ converges to a number $L$ if and only if the...
Rather than making a new post, I decided to edit this one. I tried proving the contrapositive and I feel like I have it right now. This is also suggested in the comments. I would appreciate if someone...
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Prove that $\operatorname*{\bigcup}_{i=1}^{n}[a_i,b_i]$ equals the union of...
Exercise 2.5.14 from 'The Real Numbers and Real Analysis' by Ethan D.Bloch.Hello, today I tried to prove this statement, but I'm stuck so I would like some help. Below I will put a drawing that more or...
View ArticleBounding $N$ from below to simplfy $\epsilon-N$ proofs simply [closed]
When we do the epsilon delta proof for even the simplest of functions, it is a crucial step to restrict $\delta$ in some set of values before we give the actual value of delta example.I want to make an...
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A question about the proof of the Proposition 3.30 in Real Analysis by...
This is the proof displayed in the book, and it is too brief for me to understand. The detailed difficulties are:(i) how can I deduce the equation $F'(x)=\lim _{r\to 0}\mu_F(E_r)/m(E_r)$?(ii) what is...
View ArticleConvergence of $\sum_n \frac{|\sin(n^2)|}{n}$
A problem in Makarov's Selected problems in real analysis asks to investigate the convergence of $\displaystyle \sum_n \frac{|\sin(n^2)|}{n}$I'm clueless at the moment. I can't find any good property...
View ArticleEvaluating $\sum_{k=0}^\infty \frac{1}{(nk)!} $ for integer $n$ [duplicate]
Let $n \in \Bbb N$ be an integer. We want to compute $$\sum_{k=0}^\infty \frac{1}{(nk)!} $$This clearly makes me think of the Taylor's series of $\exp$ and I know the answer is linked to $e$ but I...
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