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, . . . }.

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.

it’s supposed to be -4+45=41

step-by-step explanation:

2. angles qrt and qst both measures 156 degrees.

step-by-step explanation:  

we have been given a diagram of a circle and we are told that measure of arc qvt is 156 degrees.  

we can see from our given diagram that angle qrt and angle qst subtends to arc qvt.

since we know that angles subtended by the same arc at the circumference are always equal.

by angle subtended by same arc theorem, measure of angle qrt and angle qst will be equal to the measure of arc qvt.

therefore, angles qrt and qst both measures 156 degrees and 2nd statement is the correct choice.