The Mathematics of ECE

A sequence $\{x_i{\}}_{i=1}^{\mathrm{\infty }}$ is an association of the natural numbers to objects $x_i$.

A sequence $\{x_i{\}}_{i=1}^{\mathrm{\infty }}$ in a normed vector space $\mathcal{V}$ is said to converge if there exists a vector $x\in \mathcal{V}$ such that $$\mathrm{\forall }ϵ>0\mathrm{\exists }{N}_{ϵ}\in \mathbb{N}\mathrm{\forall }n\ge N_{ϵ}:||x_n-x||<ϵ$$ This is often written $\underset{n\to \mathrm{\infty }}{lim}{x}_n=x$ or simply $x_n\to x$, and $x$ is called the limit of the sequence.

A sequence $\{x_i{\}}_{i=1}^{\mathrm{\infty }}$ is cauchy if $$\mathrm{\forall }ϵ>0\mathrm{\exists }{N}_{ϵ}\in \mathbb{N}\mathrm{\forall }m,n\ge N_{ϵ}:||x_m-x_n||<ϵ$$

Let $\mathcal{S}$ be an ordered set (e.g. rationals, reals). For subset $\mathcal{A}\subset \mathcal{S}$ if there exists some $u\in \mathcal{S}$ such that $\mathrm{\forall }a\in \mathcal{A}:a\le u$, the subset is said to be upper bounded by $u$.

Let $\mathcal{S}$ be an ordered set (e.g. rationals, reals), and let $\mathcal{A}\subset \mathcal{S}$ be an upper bounded subset. If there exists some $s\in \mathcal{S}$ such that $s$ is an upper bound of $\mathcal{A}$ and $\mathrm{\forall }t\in \mathcal{S}:t<s⟹t$ is not an upper bound of $\mathcal{A}$, then $s$ is called the least upper bound (or supremum) of $\mathcal{A}$. $s$ is commonly denoted $sup(\mathcal{A})$.

An ordered set $\mathcal{S}$ is said to have the least-upper-bound property if all upper bounded subsets have a supremum.

The limit inferior of a sequence is defined as $$\underset{n\to \mathrm{\infty }}{liminf}(x_n)=\underset{n\to \mathrm{\infty }}{lim}(inf(\{x_m{\}}_{m=n}^{\mathrm{\infty }}))$$

The limit superior of a sequence is defined as $$\underset{n\to \mathrm{\infty }}{limsup}(x_n)=\underset{n\to \mathrm{\infty }}{lim}(sup(\{x_m{\}}_{m=n}^{\mathrm{\infty }}))$$