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Definition. If a sequence p n , n = 1,2, . . . converges to a point p in a metric space,
then this unique point p is called the limit point of the sequence and is denoted
by
li Æ• p n
.
n
5.2.7. Theorem. Let ( X ,d) be a metric space. A subset A of X is d-closed if
and only if every limit point in X of sequences of points from A also belongs to
A .
Proof. If A is d-closed, then X - A is d-open and cannot contain any limit point of
A since every one of its points has a neighborhood entirely contained in X - A . The
converse is just as easy.
Definition. Let ( X ,d) and ( Y ,d¢) be metric spaces and consider maps f i , F : X Æ Y .
We say that the sequence of maps f 1 , f 2 ,...converges pointwise to the map F if, for
every e>0 and each p
X , there is an m so that d¢(f n ( p ),F( p )) <efor all n > m. We
say that the sequence f 1 , f 2 ,...converges uniformly to F if, for every e>0 there is an
m so that d¢(f n ( p ),F( p )) <efor all n > m and all p
X . In either case, we call F the
limit function of the sequence f 1 , f 2 ,....
Notice the important difference between pointwise and uniform convergence. In
the former case, the m depends on the e and the point p, and in the latter, it depends
only on the e. This is similar to the difference between continuity and uniform con-
tinuity. A sequence of functions can converge pointwise but not uniformly (Exercise
5.2.8). The limit function of a sequence of functions, if it exists, is unique because
limits of point sequences are unique. One question that arises in the context of
sequences of functions is whether the limit function will have a property if all the
functions converging to it have this property. The answer to this question is no in
general but getting a positive answer in certain cases is precisely why the notion of
uniformly convergent is introduced. The next theorem is one example.
5.2.8. Theorem. Let ( X ,d) and ( Y ,d¢) be metric spaces and consider maps f i , F : X
Æ Y . If the maps f i are continuous and if the sequence of maps f 1 , f 2 ,...converges
uniformly to F, then F is continuous.
Proof.
This is an easy exercise. See [Lips65].
Theorem 5.2.8 is false without the hypothesis of uniform convergence.
Definition. Let ( X ,d) be a metric space. A sequence of points p n , n = 1,2,...in X is
said to be a Cauchy sequence in ( X ,d) if for every e>0 there is an m ≥ 1 so that d( p i , p j )
<efor all i,j ≥ m.
5.2.9. Theorem.
Every convergent sequence in a metric space is a Cauchy
sequence.
Proof. Let p n be a sequence that converges to a point p in a metric space ( X ,d). Let
e>0. Choose m so that n > m implies that d( p n , p ) <e/2. It follows that
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