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An attempt to compile the write up of my notes through LaTeX. Some trimming and fonts are needed in future. Testing out.

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Choosing suitable value N for equivalent sequence , such that C.seq is satisfied . (Attempt)

The aim is to show that both sequence (an)n∞ & (bn)n∞ are equivalent sequence of rationals , therefore to show that | bn - an | < ε . Then  (an)n∞ is a C.seq iff  (bn)n .

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Return progress on Tao’s Real Analysis; Chapter 5 Real number,  29 Sep

Subtopic 5.1 to 5.2 Equivalent Cauchy’s Sequence 

Firstly , the real numbers are to be constructed from the backbones of the Rationals Q , as Tao puts it in his definition that these Rationals are to be constructed with limits from a Cauchy Sequence. Thereby defining the Real from the rationals as a limit from a Cauchy sequence. 

A recap on Cauchy Sequence based on my interpretation of Tao’s definition, a few properties needed to be specified firstly for it to be one. The informal usage of ε - steadiness , a steadiness property within a given sequence. This sort of sequence can be denoted as (an)n=0 to ∞ . Sequences as you might already know are laid out as a collection of elements { 2,4,6,8,10,...} as infinite or finite {1,4,6,8,10}. To my understanding it is a way of representing a collection of numbers, terms , or elements. You might have seen Geometric sequences

{1, 1/2,1/4,1/8,1/16,....} as well or the harmonic sequence {1, 1/2,1/3,1/4,1/5,....}.

(ε steadiness) & ( ε closeness) 

The properties describes which leads up to Cauchy Sequence in his book were firstly , he mentioned the various types of sequences which can display the concept of  ε - steadiness, such as a pair of sequence given say ,  (an)n=0 to ∞  (bn)n=0 to ∞ for all indices n ≥ N and that the absolute distance between both sequence’s elements with their are to be smaller than equal to epsilon ε. However you can also do an equivalent Cauchy Sequence with two separate ones such as  (an)n=0 to ∞ and (bn)n=0 to ∞ . (This one later)

Epsilon  ε is said to be defined initially for positive rationals , however later once the Real was constructed it can be extended towards the Real , but even so the author stresses that this concept of  ε - steadiness is only temporary as its an informal way of aiding the reader . (I'm curious as to what will replace it in future sub chapters as we go on) .

Additional concept to know beforehand is also the Eventual steadiness of the sequence. 

Such that the sequence will eventually settle to an agreeable  ε - closeness of say 0.1 . Such as the sequence stated in the text { 1.41 ,1.414,1.4142,...} .

I tried to justify for myself here as to show that it eventually leads to 0.1 closeness with reference to the next and proceeding elements.

Cauchy Sequence

Therefore when  (an)n=0 to ∞  is eventually  ε  steady , for every proceeding element within such a given sequence there is a distance between them that all of them will be bounded under  ε. (The first sketch , 2nd line) .

More practice is needed. Later more interesting such as bounded sequence and equivalent C.Sequence will emerge and event to find the N value such that it falls under the satisfaction of Cauchy sequence is also elegantly thought by the author. more to come.

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Maps , directions and Relations in Hoftstader’s ASU with respect to the actual mapping of USA

In the chapter Minds and Thoughts, ASU in my understanding of Hoftstader’s analogy, a portrayal of a hypothetical created map which relates fairly well to the real maps of the USA can somehow be linked to how brains of humans give reference to objects or location in space.

In my understanding, the concept of ASU is a map whereby one can try to conjure a layout of an imaginary place, in which it can consist of as many routes, districts and even landmarks. Main roads and small interconnected ones are also permitted, since it's based on the individual's own creativity it can create as much as it can however complex it can. 

The part on correlation to minds gets interesting when in this analogy the said individual were to graph out the maps with relation to the “Real” , meaning the map of real USA, whichever region he so chooses as long with reference to it. There are global and local isomorphism mentioned for symbols and their neurons triggering ability within brains. Thereby in totality brains of humans are like the ASU analogy , each individual is quite different in terms of their route and mappings however there exists in each a main common element such as main landmark, highways, buildings etc. These are common reference point from what I understand, as the chapter puts it class categories in everyone’s brain which have in common , however when mentioning the intricate routes like in ASU, the differences here are apparent to go from one concept by symbol triggering to another would be equivalent to taking the obscure route which the individual conjure and each minor route would be different for everyone, however as long there remains a main landmark/class the destination would be the same. Main highways and landmarks are nevertheless determined by external conditions, hence even the common class categories which we identify with may be due to the external physical reality which embedded such classes of symbols in our heads in the first place.

My understanding and impression of Hoftstader’s use of symbols and class symbols is rather an effective way of conceptualizing the brain's way of recognizing and storing memory for objects perceived by the brain, evidently of course as he mentioned there are some class or symbols which behaves as absolute “knowledge” symbols. Such as concepts stemming from beliefs , religion, and cultural elements and even more so a conglomerate of symbols can even form subsystems within the brain.

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Current progress on two Literature

First literary project as for the coming Q4,2021, would be completing the “tome” like book by Douglas Hofstadter. Titled “Gödel, Escher, Bach” , very late for me to undertake this as I'm sure many have read it , nevertheless I do encourage anyone to browse it. 

The second literary was a fiction which just picked off recently at the end of Q3. Now this one is Fiction by DFW “infinite jest” I picked this because I want to see what's the fuss is all about,and this particularly I can't get heads or tails from online reviews as they seem to be in argument with one another,thus far on my own it's quite entertaining though I'm no writing expert or anything but the writing style alternates from time to time which keeps my attention from slipping away and a few good laughs here and there, not used at reading fiction hence this would be the first in many years , the last where I left off was from Nabakov’s , with guilt I never finished (maybe after this will resume). Fiction in my view is something hard to grasp. Been much more towards info based/textbook style of literature….

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Introductory, the start section of what it's about.

Content of which is likely to be found here is probably too insignificant to many eyes, the sole purpose of writing here is mainly to track my progress on how much I've learned with any given topic at hand. Mostly the ones which pique my interest. I do want to track and record the thoughts & concepts that’s been living rent free in my memory in words to which seems otherwise strange or hard for me to grasp because i'm running on a fairly low brain capacity most times, such blog then enables me to read through my thoughts at least in a comprehensive manner, and the added benefits is whoever sees it and takes interest in the concepts Im currently undertaking or have been struggling than they are more welcome to follow along my journey into understanding stuff...in general i guess