Articles of real analysis

Cardinality of separable metric spaces

Show that $\mathbb R$ is cardinally larger than any separable metric space S. I have been trying to solve this on my own. My idea was to start by mapping the open balls of positive rational radius around the points in the dense set of S to the corresponding in $\mathbb R$. Now any ${x}\subset […]

Show that $\lim_{n\to \infty} (\sqrt{n+1} – \sqrt n) = 0$ using the definition of a limit

I found that the $\lim_{x\to \infty}\left( \sqrt{n+1}-\sqrt{n}\right )=0$. But I have to use the definition of a limit to solve too. So far I have given $\epsilon>0, |\sqrt{n+1}-\sqrt{n}|< \epsilon.$ And now I don’t know what to do.

Cauchy-Schwarz for integrals

One of the exercises (3.2) of Izenman’s Modern Multivariate Statistical Techniques is for $A \subset \mathbb{R}$, $$\left( \int_{A}fg\right)^2 \leq \left(\int_{A}f^2\right) \left(\int_{A}g^2\right)$$ where $f: A \to \mathbb{R}$ and $g: A \to \mathbb{R}$ are such that $f^2$, $g^2$ are integrable. Now I’ve already proven this for a general inner product, i.e., over a vector space $V$ where […]

Is the unit sphere in $(C, \| \cdot\|_1)$ compact?

Consider the normed space $(C[0,1], \| \cdot\|_1)$ where $C[0,1]=\{f:[0,1] \to \Bbb R : f$ is continuous$\}$ and $\|f\|_1 = \int_0^1|f(t)|dt$. I’m trying to find out if the unit sphere $S=\{f \in C[0,1] : \| f \|_1 = 1\}$ is compact or not. To prove it’s not compact (I don’t know if that’s true or not) […]

“Counterexample” for a weaker version of Riemann–Lebesgue lemma

My teacher gave us this version of Riemann–Lebesgue lemma in class: Let $g(t)$ be an absolutely integrable function on $[a,b]$, then $$\lim_{p\to\infty} \int_a^b g(t)\sin(pt)dt=0$$ Similarly for $\cos(px)$ Then, he told us to look for an example of a function $g(t)$ such that it is integrable (in the improper sense) such that $$\lim_{p\to\infty} \int_a^b g(t)\sin(pt)dt\neq0$$ i.e., […]

$\lim_n \frac{b-a}{n}\sum_{k=1}^{n}\sup_{x\in}f$, $\lim_n \frac{b-a}{n}\sum_{k=1}^{n}\inf_{x\in}f$ and Darboux integrals

Let us use the notation $\overline{\int_a^b}f(x)dx$ for the Darboux upper integral of $f$ and $\underline{\int_a^b}f(x)dx$ for the lower one. Let us construct a partition of $[a,b]$ into $n$ intervals $[x_{k-1},x_k]$ defined by $x_k=a+k(b-a)/n$ and les us consider the corresponding Darboux sums$$\Delta_n=\frac{b-a}{n}\sum_{k=1}^{n}\sup_{x\in[x_{k-1},x_k]}f(x),\quad \delta_n=\frac{b-a}{n}\sum_{k=1}^{n}\inf_{x\in[x_{k-1},x_k]}f(x).$$ It is clear, by taking the definitions of $\sup$ and $\inf$, and the […]

Showing $\left \lvert \sum_{k=1}^n x_k y_k \right \rvert \le \frac{1}{\alpha} \sum_{k=1}^n x_k^2 + \frac{\alpha}{4} \sum_{k=1}^n y_k^2 $

Let $\vec x, \vec y \in \mathbb{R}^n$ and $\alpha > 0$. Show that $\left \lvert \sum_{k=1}^n x_k y_k \right \rvert \le \frac{1}{\alpha} \sum_{k=1}^n x_k^2 + \frac{\alpha}{4} \sum_{k=1}^n y_k^2 $. $$LHS = \left\lvert \vec x \cdot \vec y\right\rvert$$ \begin{align*} RHS &= \frac{1}{4 \alpha}\left(4 \sum_{k=1}^n x_k^2 + \alpha^2 \sum_{k=1}^n y_k^2\right) \\&= \frac{1}{4 \alpha}\left(2\sqrt{ \sum_{k=1}^n x_k^2} – \alpha […]

Using mean value theorem for multiple inequalities

Use the Mean Value Theorem to prove that $\frac{(x-1)}{x} < \ln x < x-1$ for $x > 1$. I was thinking of breaking up the inequality into \frac{(x-1)}{x} < \ln x$, and $\ln x < x-1$ and just proving each individual inequality using the mean value theorem approach: So, $\ln x – \frac{x-1}{x} > 0$. […]

Prove that $\lim_{x \rightarrow 0} \mathrm {sgn} \sin (\frac{1}{x})$ does not exist.

My progress: Using the sequential criterion for limits, I constructed two sequences $(x_n), (y_n)$ with $\lim(x_n)=\lim(y_n)=0$, such that $\lim(f(x_n))\neq \lim(f(y_n))$, where $f(x)=\sin\frac 1 x$. So, $\lim_{x \rightarrow 0} \sin (\frac{1}{x})$ does not exist. I also showed separately in the same way that $\lim_{x \rightarrow 0} \mathrm{sgn} (x)$ does not exist. I know that $$\lim_{x \rightarrow […]

Supremum and union

$A_i \subset \mathbb{R}, i \in \mathbb{N}$ Is $$ \sup \{\sup A_i, i \in \mathbb N\} = \sup (\cup_i A_i)?$$ Thanks.