Weierstrauss Theorem:
if a function
f
:
X
!
R
is continuous and
X
is
compact, then
f
achieves a maximum on
X:
Notation (ball): A ball with radius
"
around a point
x
is denoted as
B
(
x; "
)
:
A set
X
is bounded if, for each
x
2
X;
there
exists an
" >
0
such that
X
B
(
x; "
)
.
X
is open if, for all
x
2
X;
there exists some
" >
0
such that the open ball
B
(
x; "
)
X:
x
if, for all
" >
0
;
there exists some
n
such that
x
i
2
B
(
x
i
; "
)
for all
i > n:
Translation:
a sequence converges if it±s possible for me to keep getting
closer and closer to the limit point
x;
and once I get close (i.e. once I get
within
"
of
x;
I never move farther away.
A set
X
is closed if no sequence of points in
X
converges to a point outside of
X:
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 Fall '07
 RAUCH
 Topology, Mathematical analysis, Metric space, Topological space, Closed set, De…nition

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