Question

If = { 1 ∶ ∈ ℕ} then Inf = ⋯?
a) -1
b) 0
c) 1
d) Does not exist.​

Answer 1

Answer: Find inf S and sup S and prove your answers.

Solution We claim that inf S = 1/2 and sup S = 2. Note that, if n is

odd, 1−(−1)n/n =1+1/n, while, if n is even, 1−(−1)n/n = 1−1/n.

It follows, if n is odd, that 1 − (−1)n/n > 1 > 1/2. If n ≥ 2 is even,

1 − (−1)n/n = 1 − 1/n ≥ 1 − 1/2=1/2.

Arguing similarly, 1 − (−1)n/n ≤ 2 and so 1/2 and 2 are, respectively,

lower and upper bounds for S. Since 1/2 ∈ S, there cannot be a lower

bound m > 1/2 and so 1/2 is the greatest lower bound for S, i.e.

inf S = 1/2. Since 2 ∈ S, there cannot be a upper bound M < 2 and

so 2 is the least upper bound for S, i.e. sup S = 2.

2 Let S ⊆ R and suppose that s∗ = sup S belongs to S. If u 6∈ S,

show that

sup (S ∪ {u}) = sup{s∗

, u}.