Sequence

A sequence is a function whose domain is usually the natural numbers.

A sequence is Fejér monotone with respect to a nonempty set $C$ if $\Vert z^{k+1}-p\Vert \le \Vert z^k-p\Vert$ for every $p\in C$. It is bounded. If it has a cluster point $\bar{z}\in C$, its nonincreasing distance to $\bar{z}$ and that convergent subsequence force the entire sequence to converge to $\bar{z}$.

A finite repetition-free sequence in a set $X$ is an injective function from an initial segment $\{0,\dots ,k-1\}$ of the natural numbers into $X$, including the empty sequence for $k=0$. For a finite set of size $m$, the number of these sequences isThe order of each list is part of the data, so it can be enumerated without choosing an order on its underlying support.

A real sequence $(a_n)$ is subadditive when $a_{m+n}\le a_m+a_n$ for all positive $m,n$. Fekete lemma givesallowing the value negative infinity.

If the real error depends only on the second index and ${\alpha }_n/n\to 0$, the normalized sequence has a limit in $[-\infty ,\infty )$. For fixed $k$, write $n=qk+r$ with $1\le r\le k$ and iterate to obtain $x_n\le x_r+q(x_k+{\alpha }_k)$. It follows that $\mathrm{limsup}{x}_n/n\le (x_k+{\alpha }_k)/k$ for every $k$; a subsequence attaining the lower limit proves convergence. The limit is ${\inf }_k(x_k+{\alpha }_k)/k$, even when the errors can be negative.

A geometric progression is a sequence whose successive terms have a constant ratio: $a_n=a{r}^n$ for constants $a$ and $r$.

A subsequence of $(a_n)$ has the form $(a_{n_k})$, where $n_1<n_2<\cdots$ is a strictly increasing sequence of indices.

Every real sequence has a monotone subsequence, without a boundedness assumption. Call an index a peak when its term is at least every later term. Infinitely many peaks give a nonincreasing subsequence; beyond finitely many peaks each term has a larger later term, which recursively gives an increasing subsequence.

A null sequence is a sequence that converges to the additive identity zero in its ambient topological vector space.

A sequence $(x_n)$ in a metric space is bounded when all its terms lie in some ball of finite radius.

A bounded subsequence is a subsequence whose terms all lie in one fixed bounded set.

A binary sequence is a finite or infinite sequence whose terms belong to $\{0,1\}$.

Partition the positive natural numbers into three infinite coordinate sets. Fix the coordinates on the first two sets to any prescribed binary values, and leave those on the third set free. Reading the third coordinate set embeds all infinite binary sequences in the resulting family, so the family is an uncountable set by Cantor's diagonal argument. For example, fixing $x_{3k-2}=0$ and $x_{3k-1}=a_{3k-1}$ while leaving $x_{3k}$ free guarantees infinitely many zeros and infinitely many agreements with any prescribed binary sequence $(a_n)$. Infinite free coordinates remain even after both requirements are enforced.

A real sequence is increasing when $a_{n+1}\ge a_n$ and decreasing when $a_{n+1}\le a_n$.

A real increasing sequence bounded above converges to its supremum; a decreasing sequence bounded below converges to its infimum. For the increasing case, the definition of supremum provides a term above $L-\epsilon$, and every later term remains between that term and $L$. Negation reduces the decreasing case to the increasing case. This elementary result is distinct from the measure-theoretic monotone convergence theorem for integrals.

The limit superior of a real sequence $(a_n)$ isIt is the largest subsequential limit when the sequence is bounded.

The limit inferior of a real sequence $(a_n)$ isIt is the smallest subsequential limit when the sequence is bounded.