Question

Decide which of the following sets are compact. For those that are not compact, show how Definition 3.3.1 breaks down. In other words, give an example of a sequence contained in the given set that does not possess a subsequence converging to a limit in the set.
a.) The set of all natural numbers.

b.) The set of all rational numbers intersected with [0, 1].

c.) The Cantor set.

d.) {1 + 1/22 + 1/32 + . . . + 1/n2 : n >=1 }.

e.) {1, 1/2, 2/3, 3/4, 4/5, 5/6, . . . }.

Answer 1

Answer:

a) The set of all natural numbers not compact. he sequence (1, 2, 3, . . .) is contained in Q, but has no convergent subsequences.

b) The set of all rational numbers intersected with [0, 1] (i.e,) Q ∩ [0,1]

Not compact. Define the sequence (an) recursively by setting a1 = 1 and

an+1 = 1/4−an for all n= 1,2,3, . . ..Each an is in the set Q ∩ [0,1] but,he sequence (an) converges to 2 − √3,

which is irrational and hence not in the set Q ∩ [0,1].

c) The Cantor set: Not compact. The sequence (1,2,3, . . .) has no convergent subsequences.

d) (1+1/2.2+1/3.2+....+1/n2: n>=1):

Not compact. The sequence (1/2,1/3,1/4, . . .) consists of elements of the set, but the sequence (and, thus, any subsequence) converges to 0, which is not in the set.

e) {1, 1/2, 2/3, 3/4, 4/5, 5/6, . . . }:

The sequence {1,1/2,2/3,3/4,4/5, . . .} is compact.