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Let $1<p<\infty$ and $q=p/(p-1)$. Every $y\in {\ell }^q$ definesHölder's inequality gives $\Vert f_y\Vert \le \Vert y{\Vert }_q$, and equality follows by using a normalized sequence proportional to $\mathrm{sgn}(y_n)|y_n{|}^{q-1}$.

Conversely, for $f\in ({\ell }^p{)}^{\ast }$ put $y_n=f(e_n)$. On the first $N$ coordinates, choose the same norming vector to obtainLetting $N\to \infty$ shows $y\in {\ell }^q$. On finitely supported sequences, linearity gives $f=f_y$; density of those sequences in ${\ell }^p$ extends this identity to all $x$. Thereforeisometrically.

Yes. For $x=(x_n)\in c$, let $L={\lim }_n{x}_n$ and defineThe tail tends to zero, so $T(x)\in c_0$, and $\Vert T(x){\Vert }_{\infty }\le 2\Vert x{\Vert }_{\infty }$. Conversely, for $y=(y_n)\in c_0$, defineThis sequence converges to $y_1$ and has norm at most $2\Vert y{\Vert }_{\infty }$. The formulas are inverse bounded linear maps, so $c$ and $c_0$ are isomorphic Banach spaces.

Every bounded sequence in $\mathbb{R}$ has a convergent subsequence.

A neighbourhood of $x$ contains an open set containing $x$; $x_n\to x$ when every neighbourhood eventually contains every $x_n$. If two metrics have the same topology they plainly have the same convergent sequences. Conversely, if a $d$-open set $U$ were not $d\prime$-open, some $x\in U$ would have points $x_n\notin U$ with $d\prime (x_n,x)<1/n$; the common-sequence assumption would imply $d(x_n,x)\to 0$, contradicting $d$-openness. Symmetry finishes the proof. This fails generally: on an uncountable set the discrete and cocountable topologies differ, but in both every convergent sequence is eventually constant.

A base covers the space and refines intersections. On $X^k$, products of opens form a base. The weighted metric satisfies the metric axioms and has exactly this topology; convergence is coordinatewise. For $X^{\mathbb{N}}$ the metric $d(f,g)={\sum }_{n\ge 1}{2}^{-n}\min \{1,\rho (f(n),g(n))\}$ induces the stated product topology: finitely many coordinates control a basic neighbourhood and the tail is uniformly small. Thus sequences converge exactly coordinatewise.

Bounded means finite diameter; sequential compactness means every sequence has a convergent subsequence. A failure of uniform continuity supplies two close sequences whose image distances stay apart; a convergent subsequence contradicts continuity. Into a discrete metric, continuity is local constancy, and compactness upgrades the radii uniformly. The Cantor set is closed in compact $[0,1]$, hence sequentially compact. Uniform local constancy gives a positive separation scale, so finitely many initial Cantor-code (binary) digits determine $f$. Removing zero destroys compactness: the hinted function reading the digit after the first $1$ is continuous at every remaining sequence but depends on arbitrarily late digits.

$H$ is a closed linear subspace of reflexive $L^p$. A minimizing sequence is bounded by the triangle inequality. Reflexivity supplies a weakly convergent subsequence with limit $f_0\in H$; each norm $\Vert f-g_i{\Vert }_p$ is weakly lower semicontinuous, so their sum attains the infimum at $f_0$.

Again the series converges absolutely. $T=2{\sum }_j{B}_j/3^j$ has the Cantor distribution. Every particular digit sequence has probability zero, so it has no atoms; its support is the Cantor set, which has Lebesgue measure zero, so it cannot have a Lebesgue density.

The Rado order has domain $R=\{(m,n)\in {\mathbb{N}}^2:m<n\}$ and $(m,n){\le }_R(k,l)$ iff either $m=k$ and $n\le l$, or $n<k$. It is a well-quasi-ordering: either one row recurs infinitely in a sequence, or the first coordinates are unbounded and yield a cross-row comparison. Its infinite rows form an infinite antichain under Hoare domination preorder, showing that arbitrary-subset lifting need not preserve well-quasi-ordering.

A graph ray is an infinite sequence $v_0,v_1,\dots$ of distinct graph vertices with an edge between consecutive graph vertices. Every infinite connected locally finite graph contains a graph ray from each graph vertex, by the König infinity lemma applied to its finite self-avoiding graph paths. An open graph ray witnesses an infinite percolation cluster.

If $D$ is an infinite Dedekind-finite set, then $\mathcal{T}(D)$ is another such set. Single-vertex rooted trees inject $D$ into it. A countable sequence of distinct rooted trees gives finite lists of labels by the depth-first traversal of a tree in the prescribed child order. By countable union of explicitly ordered finite lists without choice, an infinite union of labels contradicts Dedekind-finiteness. A finite union $F$ is equally impossible, since every rooted tree then lies in the finite set $\mathcal{T}(F)$, by mathematical induction on $|F|$ using the recursive child rule. Thus no countably infinite subset of rooted trees exists.

Symbolic dynamics studies dynamical systems whose points are sequences of symbols from an alphabet and whose evolution is a shift. A finite alphabet with the discrete topology gives its sequence space a product topology.

Expand an infinite first-order structure by Skolem functions, choose distinct elements $b_n$, and fix a nonprincipal ultrafilter $\mathcal{U}$ on $\mathbb{N}$. For each finite subset $F$ of the required index order, form the ultrapower over ${\mathbb{N}}^F$ using the ordered Fubini product of ultrafilters. Pullback along coordinate projections gives coherent elementary maps by the Łoś theorem. In their directed limit of elementary embeddings, the classes of the coordinate functions $\mathbf{n}\mapsto b_{n_i}$ are distinct and form an order-indiscernible sequence; every first-order formula on increasing coordinates has the same nested ultrafilter test. Their Skolem hull gives an Ehrenfeucht-Mostowski model. An index-order automorphism sends a term in the generators to the same term in the transported generators; indiscernibility makes this well defined. Uniqueness is in the specified Skolem expansion, not necessarily in its reduct. If generators must respect a definable infinite order, first produce an increasing sequence by an ultrapower of finite increasing chains.

A two-sided sequence over a finite alphabet is uniformly recurrent if every finite word over an alphabet occurring in it occurs with bounded gaps. More precisely, for each such word some $M$ ensures that every length-$M$ interval contains a complete occurrence. This is equivalent to being a minimal point of the full shift: a finite cover of the orbit closure by preimages of a word's cylinder set bounds its return gaps; conversely, bounded gaps pass to all points of the orbit closure and make every forward orbit dense there. The property concerns finite words, not infinite integer intervals.