Sequences of real numbers
Theorem 2.1.3 (Triangular inequality) For all $x,y\in \mathbb{R}, \lvert x+y\rvert\leq \vert x\rvert+\vert y\rvert$
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Countable and Uncountable sets
Definition 1.7.2 For each positive integer $n\in \mathbb{N}$, let \(\mathbb{N}_n:=\{1,2,\ldots,n \}\). If $A$ is a set, we say (a) $A$ is finite if $A\sim \...
Least Upper Bound Property
Definition (Ordered Set) Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following property
Set and Function
Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$