Basic properties of Metric Spaces (from Mathematical Analsys 2 by Zorich)

Proposition 1:

Let $(X,d)$ be a metric space. a. The union $\bigcup_{\alpha\in{A}}G_{\alpha}$ of the sets in any system $\set{G_{\alpha}, \alpha\in{A}}$ of sets that are open in $X$ is and open set in $X$. b. The intersection $\bigcap_{i=1}^nG_i$ of any finite number of sets that are open in $X$ is an open set in $X$. c. The intesection $\bigcap_{\alpha\in{A}}F_{\alpha}$ of the sets in any system $\set{F_{\alpha}, \alpha\in{A}}$ of sets $F_{\alpha}$ that are closed in $X$ is a closed set in $X$. d. The union $\bigcup_{i=1}^nF_i$ of any finite number of sets that are closed in $X$ is a closed set in $X$.

A proof of the Euclidean Algorithm

The Euclidean Algorithm:

Let $m,n\in\mathbb{N}$ such that $m>n$. Then we define succesively -

$$ m = q_1n+r_1 \ n = q_2r_1+r_2 \ r_1 = q_3r_2+r_3 \ .. \ r_k = q_{k+2}r_{k+1} + 0 $$

than $\gcd(m,n)=r_{k+1}$

Initial (wrong) proof: If $\gcd(m,n)=1$ than $m,n$ are foreign (or mutually prime). Otherwise, assume $\gcd(m,n)>1$. Define $r_0=n$ We prove by induction that $r_{i+1}|r_i$.

Base case - If $r_1\nmid{n}$ than we can deduce that $\gcd(m,n)=1$. Contradiction.

A question on the expansion of rational numbers in base-q

Question: A number $x\in{\mathbb{R}}$ has a periodic expansion (in any number system) $\iff$ $x$ is rational.

Proof:

$\impliedby$ Assume $x$ is rational, i.e. there are $m,n\in{\mathbb{Z}}$ such that, $x=\frac{m}{n}$ and $n\neq0$ - Assume this is the most reduced form. Than, by way of long division we obtain deteministically the decimals of its expansion in some $q$-ary number system:

We define $r_0$ to be the remainder of $m$ upon division by $n$. We define the following recursive relation obtained from the “long division” process -

A question from the 2025 UBC qualifying exam

Let $f:(0,\infty)\rightarrow\mathbb{R}$ be a twice differentiable function. Further suppose There are constants $A,B,C$ such that -

$A=\sup_{x>0}{|f(x)|}\newline B=\sup_{x>0}{|f’(x)|}\newline C=\sup_{x>0}{|f’’(x)|}$

Prove that $B^2\leq{4AC}$:

a. Use Taylor’s Theorem to expand $f(x)$ about $a$ and compute $f(a + 2h)$ for $h > 0$.

b. Rearrange the resulting expression to isolate $|f(a)|$ and bound it in terms of $A$, $C$ and $h$.

c. Now pick $h$ to minimise your bound on $|f(a)|$ and clean up.

A proof of Rolles Theorem and Lagranges finite increment Theorem

Theorem: A continous function on a closed interval $[\alpha,\beta]$ attains a maximum or a minimum in the closed interval.

Proof: From the definition of continuity - for any point in $[\alpha,\beta]$ there exists a neighborhood such that the function is bounded in that environment.

[Side note: For that reason the function $f(x)=\frac{1}{x}$ is not continous on any interval containing 0, since it is unbounded at any environment of the point.]

The number of binary strings on $n$ letters with no two consequitive ones

This question is from a lecture by Prof. Gil Cohen that popped on my YT feed (To my discredit, I watch YT on my free time..).

Question: “How many binary strings on $n$ letters are there with no two consequitive ones?”

Observations:

1. Naivly, For any $n$, if the number of 1s is strictly larger than $n/2$ than by the pigenhole principle we surely have two consequitive ones.
2. But ofcourse we can have consequitive 1s when the number of 1s is strictly lower than $\frac{n}{2}$. (e.g. 01100).

Some simple examples:

Yet another limit pf from Ash 😅

Problem: Suppose $\mu$ is a probability measure (or nonegative countably additive set function) and the following holds - $A_i\uparrow A$ and $A_i’\uparrow A’$ such that $A\subset{A’}$ than we have - $\lim_i\mu(A_i)\leq\lim_i\mu(A_i’)$.

The simplest proof possible: As we know from previous statements on countably additive set functions - $$\lim_i\mu(A_i)=\mu(\lim_i{A_i})=\mu(A)\leq\mu(A’)=\mu(\lim_i{A_i’})=\lim_i\mu(A_i’)$$

A first attempt of a proof using elementary analysis: Naively, we may try to attack this question as purely a question on sequences of real numbers $\set{\mu(A_i)}_i$ and $\set{\mu(A_i’)}_i$?

A question on the cardinality of a $\sigma$-field

Let $\gamma$ be a any class of subsets of $\Omega$ such that $\phi,\Omega\in\gamma$. We define $\gamma_0=\gamma$, and for any ordinal $\alpha>0$, inductively:

$$\gamma_\alpha = (\cup\set{\gamma_\beta:\beta<\alpha})’$$

Where $D’$ is the class of all countable unions of differences of sets in $D$.

What is $D’$? Assume $D = \set{\Omega, \phi}$ The difference $\Omega-\phi$ is the set $\Omega$. Assume $A\in{D}$ than $D’$ contain $\phi-A$, $\Omega-A=A^c$, $\phi-A=\phi$, etc.

But than - $A=\Omega-A^c\in{D’}$!

We than define $\frak{U}=\cup\set{\gamma_{\alpha}:\alpha<\beta}$

A proof for $\mu(\lim_{n\rightarrow\infty}{A})=\lim_{n\rightarrow\infty}{\mu(A_n)}$

Definition: If $\liminf A_n = \limsup A_n = A^{\star}$ we define $\lim_n A_n = A^{\star}$.

Problem: Assume $\mu$ a finite measure and $A_n\in\mathcal{F}, \forall{n}\in\mathbb{N}$. Show $\mu(\lim_{n\rightarrow\infty}{A})=\lim_{n\rightarrow\infty}{\mu(A_n)}$

An initial idea:

My initial idea was to use the following properties of $\liminf A_n$ and $\limsup A_n$:

1. $w\in\liminf{A_n} \iff \exist{n}$ such that $\forall{k\geq{n}}$ we have $w\in{A_k}$.

2. $w\in\limsup{A_n} \iff$ there are infinitely many $n$ such that $w\in{A_n}$.

However these propoerties do not reveal a connection between $\mu(\limsup{A_n})=\mu(\liminf{A_n})=\mu(A^{\star})$ and $\mu(A_n)$.