[Lecture Summary] 00 Real Anaysis : TOC

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Theorem

1. De Morgan’s Law
2. For $c\ge -1 \mbox{ and } n\in \mathbb{N}, (1+c)^n \ge 1+nc$
3. Cantor-Schroder-Bernstein
   • $\mid A \mid \ge \mid B \mid$ and $\mid A \mid \le \mid B \mid$ then $\mid A \mid = \mid B \mid$
4. Archimedian Property
   • It is useful when it comes to prove ineqaulity in other theorems.
5. Density of $\mathbb{Q}$
6. Squeeze theorem!
7. Reverse triangle inequality
   • $\lvert \lvert a\rvert - \lvert b\rvert\rvert \le \lvert a-b\rvert$
8. If one of elements which belongs to a set is euqal to the upper bound, and it is a supremum!
9. Bolzano-Weierstrass
   • Every bounded sequence has a convergent subsequence.
   • Once we prove a boundedness of sequence, and it is natural to assume that subsequence of that set is convergent!

AXIOM

1. Well-ordering propery of natural numbers
   • There is always the smallest element

Concepts

	$\mathbb{Z}$	$\mathbb{N}$	$\mathbb{Q}$	$\mathbb{R}$
LUBP			NO	YES
	commutative ring	Ordered Field	Countable

Flow

1. upper bound & LUBP + Field!
2. $x$, Supremum of a given set
   • upper bound for that given set
   • find an element which belongs to that given set and exists between $(x-\epsilon,x]$
3. proof of convergence(limits)
   • Choose $M$ with some condition.
   • Then show that $M$ works(derive inequality based on that condition and definition)
   • Or Squeeze Theorem.
   • show that is is bounded above and below and both bound converges to the same value! - $\lim \lvert x_n - x \rvert = 0$
4. monotonic increasing & bounded implies convergence!
   • Series if monotonic increasing for natural number!