Cauchy Sequence

Definition: Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. It can be defined for various spaces, such as:

For real numbers, a sequence $\langle x_n\rangle$ of real numbers, such that:

$\qquad\quad\langle x_n\rangle = x_1, x_2, x_3, \cdots$ where, $x_1, x_2, x_3, \cdots\;\in\;\mathbb{R}$

is a Cauchy sequence, if for every positive real number $\epsilon$, $\exists$  an $N\;\in\;\mathbb{N}$, such that

$\qquad\quad |x_n - x_m| < \epsilon \;\;\forall\;\; n, m\;\geq\; N$

Similarly, for rational numbers, a sequence $\langle x_n\rangle$ of rational numbers, such that:

$\qquad\quad\langle x_n\rangle = x_1, x_2, x_3, \cdots$ where, $x_1, x_2, x_3, \cdots\;\in\;\mathbb{Q}$

is a Cauchy sequence, if for every positive rational number $\epsilon$, $\exists$  an $N\;\in\;\mathbb{N}$, such that

$\qquad\quad |x_n - x_m| < \epsilon \;\;\forall\;\; n, m\;\geq\; N$


For example:

$\textrm{If}\;\;\;\quad \langle x_n\rangle =\left\{\dfrac{1}{n}\;\big|\;\; n\;\in\;\mathbb{Z}\right\}\\ \begin{align}\textrm{Then,}\; |x_n - x_m| &= \left| \dfrac{1}{n} - \dfrac{1}{m}\right|\\[6pt] & < \dfrac{1}{n} \text{&} \dfrac{1}{m}\end{align}$

Choosing:
$\begin{align}\qquad\quad & N > \dfrac{1}{\epsilon}\\ \Rightarrow\quad\;\; &\dfrac{1}{N} < \epsilon\\ \therefore\quad\;\;\; &|x_n - x_m|\;\;< \; \dfrac{1}{n} \text{&} \dfrac{1}{m} < \;\epsilon\;\;\forall\;\; n, m\;\geq\; N \;\left(\textrm{for}\; N > \dfrac{1}{\epsilon}\right) \end{align}$

Hence, $\langle x_n\rangle =\left\{\dfrac{1}{n}\;\big|\;\; n\;\in\;\mathbb{Z}\right\}$ is a Cauchy sequence.

Abelian Group

Definition: An Abelian group, or a commutative group, is a group which along with the group axioms, satisfies an additional axiom of commutativity, i.e., if $(G, •)$ is an abelian group, then it satisfies the following five axioms:

Closure:
$\forall\;\;a, b\;\in\; G, \; a • b \;\in\; G$.

Associativity:
$\forall\;\;a, b$ and $c\;\in\; G, \; (a • b) • c = a • (b • c)$.

Existence of identity:
$\exists\;$ an element $e\;\in\; G$, such that $\forall\; a\;\in\; G,$

$\qquad\quad e • a = a • e = a$

Such $e$ is known as the identity element of $G$ w.r.t. $•$.

Existence of inverse:
For each $a\;\in\; G$, $\exists$ an element $b\;\in\; G$, such that,

$\qquad\quad a • b = b • a = e$

Where $e$ is the identity element, and $b$ is called the inverse of $a$ in $(G, •)$.

Commutativity:
$\forall\;\;a, b\;\in\; G$,

$\qquad\quad a • b = b • a$

Group

Definition: A group is a set, $G$, together with an operation $•$ that combines any two elements $a$ and b to form another element, denoted by $a • b$ or $ab$.  A group $(G, •)$, must satisfy four requirements known as the group axioms:

Closure:
$\forall\;\;a, b\;\in\; G, \; a • b \;\in\; G$.

Associativity:
$\forall\;\;a, b$ and $c\;\in\; G, \; (a • b) • c = a • (b • c)$.

Existence of identity:
$\exists\;$ an element $e\;\in\; G$, such that $\forall\; a\;\in\; G,$

$\qquad\quad e • a = a • e = a$

Such $e$ is known as the identity element of $G$ w.r.t. $•$.

Existence of inverse:
For each $a\;\in\; G$, $\exists$ an element $b\;\in\; G$, such that,

$\qquad\quad a • b = b • a = e$

Where $e$ is the identity element, and $b$ is called the inverse of $a$ in $(G, •)$.

Rational Numbers

Definition: a rational number is any number that can be written in the p/q form of two integers, p and q, with the denominator q not equal to zero. The set of all rational numbers is denoted by $\mathbb{Q}$, i.e.,

$\qquad\quad\mathbb{Q} = \left\{ \dfrac{p}{q} \;\big|\;\; p, q\;\in\;\mathbb{Z};\;\; q\neq 0\right\}$

In decimal representation, a number is a rational number if and only if the decimal expansion of a number either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. This holds true irrespective of the base, i.e, decimal, binary, etc.

Related:
Integers (definition)

Integers

Definition: An integer is a number that can be written without a fractionl component. The set of integers is denoted by $\mathbb{Z}$. Further, if $n$ is an integer, then $n^+$ (or the successor of $n$, eg. $0^+ = 1$, $1^+ = 2$) is also an integer. Similarly, if $n$ is an integer, then $n^-$ is also an integer.

Median Through Hypotenuse

Theorem: In a right triangle, length of a median drawn through the vertex having right angle to meet hypotenuse, is equal to one half of the length of the hypotenuse.

Prerequisites:
Median (definition)
Midpoint Theorem (proof)
SAS congruence (proof)
Angle on a straight line (proof)
Corresponding angles property (proof)

Proof:

Let $\triangle ABC$ be a right triangle, right angled at B. Let BD be a median drawn from B to meet AC at D.

We need to show that BD = $\dfrac{1}{2}$ AC. For this, let us join DE, where E is the midpoint of AB.

Since, D and E are the midpoints of AC and AB respectively (see definition of median), hence by Midpoint Theorem,

$\begin{align}\qquad\quad &DE \parallel BC\\ \Rightarrow\quad\;\; &\angle AED = \angle ABC = 90^o\quad\qquad\qquad\!\! && \text{(corresponding angles)} &&& \cdots\text{(1)}\\ \Rightarrow\quad\;\; & \angle BED = 180^o -\angle AED = 90^o && \text{(angle on a straight line)} &&& \cdots\text{(2)}\end{align}$

Now, consider $\triangle AED$ and $\triangle BED$:

$\begin{align}\qquad\quad\;\; & AE = BE\qquad\qquad\qquad\qquad\qquad && \text{(by construction)}\\ & DE = DE && \text{(common)}\\ & \angle AED = \angle BED = 90^o && \text{(from $(1)$ and $(2)$)}\\ \therefore\quad\;\;\;\;\; & \triangle AED\cong\triangle BED && \text{(by SAS congruence)}\\ \\[12pt] \Rightarrow\;\quad\quad & AD = BD && \text{(CPCTC)}\qquad\qquad\qquad\qquad\qquad\cdots\text{(3)}\\ \text{But, }\;\quad & AD = AC/2 && \text{(by definition of median)}\\ \text{Hence, }\; & BD = AC/2 && \text{(from $(3)$)}\end{align}$

Q.E.D.

Recommended;
Location of centroid
Pythagoras Theorem
Angle Bisector Theorem

Centroid

Theorem: All the three medians of a triangle intersect at a single point called centroid, which divides each median in a ratio of $2:1$.

Prerequisites:
Similarity of triangles (definition)
Midpoint Theorem (proof)
AA similarity (proof)
Alternate angles property (proof)
vertical angle theorem (proof)

Proof:

Let ABC be a triangle having medians AD, BE and CF.
Let us consider the medians BE and CF, which intersect at a point G. Let us join E and F by a straight line.

Since EF joins the midpoints of the lines AB and AC, hence by midpoint theorem:

$\qquad\quad EF \parallel BC$
And, $\;\;\;\! EF = \dfrac{1}{2} BC\qquad\qquad\qquad\cdots\text{(1)}$

Now, consider triangles $\triangle BCG$ and $\triangle EFG$:

$\qquad\quad\angle EGF = \angle BGC\qquad\qquad\qquad\text{(vertically opposite angles)}\\ \qquad\quad\angle GFE = \angle GCB\qquad\qquad\qquad\text{(alternate angles)}\\ \therefore\quad\;\;\triangle BCG\sim\triangle EFG\qquad\qquad\qquad\text{(AA similarity)}$

Thus, by definition of similarity, corresponding sides are proportional, hence:

$\qquad\quad\dfrac{GE}{GB} = \dfrac{GF}{GC} =\dfrac{EF}{BC} = \dfrac{1}{2}\qquad\qquad\text{(from $(1)$)}$

Thus, G divides BE and CF in the ratio $2:1$.

Similarly, by considering the medians AD and BE, which intersect at a point G', it can be shown that G' divides AD and BE in the ratio 2:1.

But BE is divided in $2:1$ ratio by G. Hence, G' $=$ G.

Thus, all the medians intersect at a single point, which divides the medians in a ratio $2:1$.

Q.E.D.


Recommended;
Median through hypotenuse
Basic Proportionality Theorem
Angle Bisector Theorem