Articles of real analysis

series and inequality

I found this homework in an old paper written Let $n\in \mathbb{N}^*$ and $x_1,x_2,\ldots,x_n \in \mathbb{R}$ such that $ \sum_{i=1}^{n}\left|x_{i}\right|=1$ and $\sum_{i=1}^{n}x_{i}=0$ Show that : $\forall i \in [1;n] \quad \left|\dfrac{2}{i}-1-\dfrac{1}{n}\right|\leq 1-\dfrac{1}{n}$ Deduce that : $\left|\sum\limits_{i=1}^n\dfrac{x_i}{i}\right|\leq\dfrac{1}{2}\left(1-\dfrac{1}{n}\right)$ I tried, we’ve $ \forall i \in [1,n],$ \begin{align*} 1&\leq i\leq n \\ \dfrac{1}{n}&\leq \dfrac{1}{i}\leq 1\\ \dfrac{2}{n}&\leq \dfrac{2}{i}\leq 2\\ […]

Continuity of inverse function (via sequences)

If $f:[a,b] \rightarrow [c,d]$ is a continuous bijective function, then prove $f^{-1} $ (its inverse function) is continuous on $[c,d]$. I know this can be proven by using monotony of $f$, can anyone help me finish my approach? $\textbf{My approach: }$ Let $s_n \rightarrow s $ on $[c,d]$. And let $ t_n = f^{-1}(s_n),$ and […]

Are there conditions for which $\int_{a}^{x} O(f(t)) dt = O\left( \int_{a}^{x} f(t)) dt \right)$ (same for derivatives)

I was wondering if in asymptotics there some conditions for integral/derivatives where I can do something like $$\int_{a}^{x} O(f(t)) dt = O\left( \int_{a}^{x} f(t)) dt \right)$$ Or something like $$ \frac{d O(f(x))}{dx} = O\left( \frac{df(x)}{dx} \right)$$ I’ve been watching some videolectures, and sometimes I just see people using as nothing this stuff, but I wonder […]

Frullani identity: justifying the application of Fubini theorem

Let \begin{align} f \in C^1 \text{ with } f(0) , f(\infty) \in \Bbb R \text{ and } b>a>0 \tag{1}. \end{align} Show \begin{align*} \int_0^\infty \frac{f(ax)-f(bx)}{x}dx = \left( f(0) – f(\infty) \right) \log\left( \frac{b}{a} \right). \end{align*} There is a proof of this identity in this answer. How do we justify the application of Fubini theorem to interchange […]

How to reduce this to Sturm-Liouville form?

I have the ODE $$-(1-x^2) \frac{d^2 f(x)}{dx^2} + x \frac{df(x)}{dx}+g(x)f(x)=\lambda f(x)$$ and I want to reduce it to Sturm-liouville form. The problem is that we don’t have $2x$ but just $x$. otherwise it would be similar to the Legendre differential equation. Could anybody help me with that? By the way, does this mean that the […]

Proving $n^\epsilon > \log(n)$ for sufficiently large $n$

I’d like to show that for all $\epsilon > 0$, there exists an $N$ such that for all $n \geq N$ the following holds: $n^\epsilon > \log(n)$. I’m having trouble justifying this. My intuition says this should hold. Writing $n = 2^k$ gives an inequality of the form $(2^\epsilon)^k > k$. This should hold for […]

Definition of a separable metric space

The book I’m reading doesn’t explicitly give a definition of separable metric spaces. The only type of separability definition I know that a separable topological space is one that has a countable dense subset. Could someone give me a definition of a separable metric space? I’m assuming it would have something to do with the […]

Is $f(x,y(x))$ a function of two variables? If so, how is that possible?

I don’t know how to interpret this function: Let $D$ be an open subset of $\mathbb{R}^2$ with a continuous function $f:D \rightarrow \mathbb{R}$ and \begin{align} y'(x)=f(x,y(x)) \end{align} is a continuous, explicit first-order differential equation defined on $D$. From $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ I interpret $f$ as a two variable function, $f(x,y)$. But here $y$ is a […]

Might proof by contradiction be needed for this mundane problem?

This present question is inspired by this earlier question. Consider the problem of proving that if $f\ge 0$ on an interval $I\subseteq\mathbb R$ and $\int_I f=0$, then $f$ is $0$ (almost) everywhere on $I$. Can it be proved that the only way to prove this is by contradiction? PS: How about if we consider two […]

A perfect set results by removing the intervals in such a way as to create no isolated points.

Here is a statement in Zygmund’s Measure and Integral that confuses me on Page 8: Any closed set in $\mathbb{R}^1$ can be constructed by deleting a countable number of open disjoint intervals from $\mathbb{R}^1$. A perfect set results by removing the intervals in such a way as to create no isolated points; thus, we would […]