PROOF. Suppose on the contrary that{xn}converges to distinct pointsx and
y. The numberm = d(x,y)is then>0. By the assumed convergence,xnlies in both open ballsB(m2;x)andB(m2;y)ifnis large enough. Thusxnlies in the intersection of these balls. But this intersection is empty, since the presence of a pointzin both balls would mean thatd(x,y)≤d(x,z)+d(z,y) < m2+m2 =m,
4. Sequences and Convergence 99 If a sequence{xn}in a metric space(X,d)converges tox, we shall callx the
limitof the sequence and write limn→∞xn=x or limnxn =xor limxn=x or
xn →x. A sequence has at most one limit, by Proposition 2.20. If the definition of convergence is extended to pseudometric spaces, then sequences need not have unique limits.
Let us identify convergent sequences in some of the examples of metric spaces in Section 1.
EXAMPLES OF CONVERGENCE IN METRIC SPACES.
(0) The real line. OnRwith the usual metric, the convergent sequences are the sequences convergent in the usual sense of Section I.1.
(1) Euclidean spaceRn. Here the metric is given by
d(x,y)=≥ n X k=1 (xk−yk)2 ¥1/2
ifx=(x1, . . . ,xn)andy =(y1, . . . ,yn). Another metricd0(x,y)is given by
d0(x,y)= max
1≤k≤n|xk−yk|, and we readily check that
d0(x,y)≤d(x,y)≤pn d0(x,y).
From this inequality it follows that the convergent sequences in(Rn,d)are the same as the convergent sequences in(Rn,d0). On the other hand, the definition ofd0 as a maximum means that we have convergence in(Rn,d0)if and only if we have ordinary convergence in each entry. Thus convergence of a sequence of vectors in(Rn,d)means convergence in thekthentry for allkwith 1≤k≤n.
(2) Complex Euclidean spaceCn. As a metric space,Cn gets identified with
R2n. Thus a sequence of vectors inCnconverges if and only if it converges entry by entry.
(3) Extended real lineR∗. Here the metric is given byd(x,y)= |f(x)−f(y)| with f(x)= x/(1+ |x|)ifx is inR, f(−∞)= −1, and f(+∞) = +1. We saw in Section 1 that the intersections withR of the open balls of R∗ are the open intervals inR. Thus convergence of a sequence inR∗to a point x inR means that the sequence is eventually in(−∞,+∞)and thereafter is an ordinary convergent sequence inR. Convergence to+∞of a sequence{xn}means that for each real numberM, there is an integer N such thatxn ∏M whenevern ∏ N. Convergence to−∞is analogous.
(4) Bounded scalar-valued functions onSin the uniform metric. A sequence {fn}inB(S)converges in the uniform metric onB(S)if and only if{fn}converges uniformly, in the sense below, to some member f of B(S). The definition of uniform convergencehere is the natural generalization of the one in Section I.3: {fn}converges to f uniformly if for each≤ >0, there is an integerNsuch that
n ∏ N implies|fn(s)− f(s)|< ≤for allssimultaneously. An important fact in this case is that the sequence{fn}isuniformly bounded, i.e., that there exists a real numberM such that|fn(s)| ≤ M for alln ands. In fact, choose some integerNfor≤=1. Then the triangle inequality gives
|fn(s)| ≤ |fn(s)− f(s)| + |f(s)− fN(s)| + |fN(s)| ≤2+ |fN(s)| for allsifn∏N, so that Mcan be taken to be max1≤n≤N©sups∈S|fn(s)|™+2.
(5) Bounded functions fromSinto a metric space(R, ρ). Convergence here is the expected generalization of uniform convergence: {fn}converges to f uniformly if for each≤ > 0, there is an integer N such that n ∏ N implies ρ(fn(s), f(s)) < ≤ for all s simultaneously. As in Example 4, a uniformly convergent sequence of bounded functions isuniformly boundedin the sense thatρ(fn(s),r0)≤Mfor allnands,Mbeing some real number. Herer0is any
fixed member ofR.
(7) Indiscrete spaceX. The functiond(x,y)in this case is a pseudometric, not a metric, unlessX has only one point. Every sequence in Xconverges to every point inX.
(8) Discrete metric. Convergence of a sequence{xn}in a space X with the discrete metric means that{xn}is eventually constant.
(11) Hilbert cube. For eachn, let({xm}∞m=1)nbe a member of the Hilbert cube, and writexmnfor themthterm of thenthsequence. Asnvaries, the sequence of sequences converges if and only if limnxmn exists for eachm.
(12)L1metric on Riemann integrable functions. The functiond(f,g)defined
in this case is a pseudometric, not a metric. Convergence in the corresponding metric space as in Proposition 2.12 therefore really means a certain kind of con- vergence of equivalence classes: If{fn}and f are given, the sequence of classes
{[fn]}converges to the class [f] if and only if limnRab|fn(x)− f(x)|d x = 0. The use of classes in the notation is rather cumbersome and not very helpful, and consequently it is common practice to treat theL1space as a metric space and to
work with its members as if they were functions rather than equivalence classes. We return to this point in Chapter V.
Let us elaborate a little on Examples 4 and 5, concerning the space B(S) of bounded scalar-valued functions on a set S or, more generally, the space of bounded functions from S into a metric space (R, ρ). Suppose that S has
4. Sequences and Convergence 101 the additional structure of a metric space(S,d). We letC(S)be the subset of B(S)consisting of bounded continuous functions onS, and we writeC(S,R)or C(S,C)if we want to be explicit about the range. More generally we consider the space of bounded continuous functions fromSinto the metric spaceR. All of these are metric spaces in their own right.