“real: chapter_04” — 2011/5/22 — 23:17 — page 12 — #12
4-12 Real Analysis
Consider the open intervals (a − , a + ), (a, a + 2), (a + , a +
3), ..., (a+(n−2), a+n), which form a finite -net covering [a, b]
(note that (a−, a+)covers [a, a+), (a, a+2) covers [a +, a+2),
etc. and finally (a + (n − 2), a + n) covers [a + (n − 1), b])
and
hence E. This shows that E is totally bounded.
Summarizing the properties of closed and bounded sets in R
described by the previous theorems, we have the following.
Theorem 4.2.25 A subset E of
R is closed and bounded if and only
if it is complete and totally bounded.
Closed and bounded subsets of
R can be characterized using only the
notion of open sets. More specifically, we shall show that a set K ⊂
R
is