## Calculus: Theorems about continuous functions

** — 1. Theorems about continuous functions — **

** — 1.1. Infimum, supremum — **

**Definition 1 (least upper bound, greatest lower bound).**

*A set of real numbers is bounded above if there exists a number such that for all . Such is called an upper bound of .*

A number is callead **least upper bound** of if is an upper bound of and for all upper bound of .

Lower bound and greatest lower bound are defined similarly.

Before introducing the theorems, it is necessary to note that the set of real numbers has an important property, call least-upper-bound property, as this will be used in the proofs.

**Theorem 2 (least-upper-bound property of ).**

*For every nonempty subset that is bounded above has a least upper bound, i.e. is in .*

## Calculus: Limits

My writing about Calculus when I was/am taking MATH1051, MATH2400 and reading Spivak Calculus book.

** — 1. Definition of limits — **

** — 1.1. Definition of limits of function — **

For me, the precise definition of limits is a bit hard to understand so I decided to reconstruct the whole thing by writing some of my guesses on why did mathematicians write the definition in this way. By the way, French mathematician, Augustin-Louis Cauchy (1789-1857) is the one who introduced into definition of limits.

Limit, the notation is understood vaguely as: as gets closer to , gets closer to a number . What does it mean by ‘get closer’? It is easier to understand if we imagine lies on a number line and suppose we pick an arbitrary point on the line. If we apply the term ‘ gets closer to ‘, we can understand it as distance from to on the number line decreases. Its distance is so saying gets closer to is the same as saying gets closer to . Note that for any distance , there are two possible on the number line that can satisfy that distance. What we care about is the distance from to not the position of , so using ‘ gets closer to ‘ is better than ‘ gets closer to ‘. Similarly to and we get a new definition:

Note that we can get as close to as we want by choosing appropriate , but we can’t guarantee the same thing for since depends on . Thus, firsly, in order for to be able to approach we need to be as small as we want. It is equivalent to saying:

For any there exists some so .

Secondly, we need the definition to say approaches as approaches . This means for small enough , must be bounded by some for all . For better understanding more about this argument, let’s assume the contrary, then as gets closer to , still does not have a bound for all . Say we find at has . We consider all in and follow that there must be an so and according to the assumption. Next consider all in and from our assumption, there exists so for . This keeps going and we can see that as gets closer to , gets larger, which contradicts to our aim that must approaches . Thus, we got the second part of the definition: (more…)

## Reading a solution using algebraic integers of a HMMT number theory problem

I tried to attack the following problem but eventually gave up.

**Problem 1**

*(HMMT Ferb 2015 Team P9) Let and . Prove that*

*is an integer and find its remainder upon division by *

The only idea I have is to expand this expression, but it was too complex to do. So I decided to read the solution using algebraic integers. I’m new at algebraic number theory and I don’t know much about algebraic integers. My only reference is Problems from the book (PFTB), Chapter 9.

## Yellowstone Permutation – St. Petersburg MO 2015, grade 9, 2nd round

This problem was given to the St. Petersburg Olympiad 2015, 2nd round, grade 9 and is known as Yellowstone Permutation. You can see here for the article about this sequence and here for the OEIS entry.

**Problem. **(Yellowstone Permutation) A sequence of integers is defined as follows: and for , is the smallest integer not occurring earlier, which is relatively prime to but not relatively prime to Prove that every natural number occurs exactly once in this sequence.

*M. Ivanov*

Here is my proof for this interesting problem. Another proof can be found in the article given link above.

## The normal approximation to the binomial distribution

A coin is flipped times with probability of getting heads is . This is a binomial approximation . The Year 12 MATH B textbook gives an normal approximation to this: where is the mean (or expected value) and is the standard deviation. Since the textbook doesn’t give a proof for this so I will go and prove.

## Bài hình học Tuần 4 tháng 12 – 2015 của thầy Trần Quang Hùng

Thầy Trần Quang Hùng có một chuyên mục Mỗi tuần một bài toán trên blog của thầy. Dưới đây là bài toán của thầy Hùng đưa lên trong Tuần 4 tháng 12.

*PROBLEM. Cho tam giác nhọn với đường cao và tâm ngoại tiếp . Đường thẳng qua vuông góc với lần lượt cắt tại . Gọi theo thứ tự là trực tâm của tam giác và và theo thứ tự là trực tâm tam giác . Chứng minh rằng đồng quy.*

Sau đây là lời giải của tôi được đưa lên trong topic Mỗi tuần một bài toán trên DDTH.