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.
