The Mathematics of ECE

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

A sequence \(\{x_i\}_{i=1}^\infty\) in a normed vector space \(\calV\) is said to converge if there exists a vector \(x \in \calV\) such that \[ \forall \epsilon > 0 \exists N_\epsilon \in \bbN \forall n \geq N_\epsilon :~ ||x_n - x|| < \epsilon \] This is often written \(\lim_{n \to \infty} x_n = x\) or simply \(x_n \to x\), and \(x\) is called the limit of the sequence.

A sequence \(\{x_i\}_{i=1}^\infty\) is cauchy if \[ \forall \epsilon > 0 \exists N_\epsilon \in \bbN \forall m,n \geq N_\epsilon :~ ||x_m - x_n|| < \epsilon \]

Let \(\calS\) be an ordered set (e.g. rationals, reals). For subset \(\calA \subset \calS\) if there exists some \(u \in \calS\) such that \(\forall a \in \calA :~ a \leq u\), the subset is said to be upper bounded by \(u\).

Let \(\calS\) be an ordered set (e.g. rationals, reals), and let \(\calA \subset \calS\) be an upper bounded subset. If there exists some \(s \in \calS\) such that \(s\) is an upper bound of \(\calA\) and \(\forall t \in \calS :~ t < s \implies t\) is not an upper bound of \(\calA\), then \(s\) is called the least upper bound (or supremum) of \(\calA\). \(s\) is commonly denoted \(\sup(\calA)\).

An ordered set \(\calS\) 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 \[ \underset{n \to \infty}{\lim \inf}~ (x_n) = \lim_{n \to \infty} \bigg(\inf\big(\{x_m\}_{m=n}^\infty\big)\bigg)\]

The limit superior of a sequence is defined as \[ \underset{n \to \infty}{\lim \sup}~ (x_n) = \lim_{n \to \infty} \bigg(\sup\big(\{x_m\}_{m=n}^\infty\big)\bigg)\]