Sequences and Sets

Question

Show that {(1 + 1/n , 2 + 1/n) : n ∈ N} is an open covering of the closed interval (1,2] and

determine if it can be reduced to a finite cover. Justify your answer.

Answer 1

It is an open cover of the interval (1,2] since we can easily check that (union over all n in N of the intervals (1+1/n, 2+1/n))=(1,2]. It can not be reduced to a finite cover because the interval (1,2] is not a compact subset of the real line.