Question

Applying Bolzano-Weierstrass theorem show that the set
S={1+1/ n ∨ n∈N}∪{−1−1/ n ∨ n∈N}
must have a limit.

Answer 1

{1+1/ n ∨ n∈N}

x₁=1+1/1=2

x₂=1+1/2=1.5

1.5-2=-0.5

because of x₂<x₁ the sequence monotonously decreases

Since for all n,xn>0, $\lim _{n\to \infty }\left(1+\frac{1}{n}\right)=1$ then it is bounded below

{−1−1/ n ∨ n∈N}  

x₁=-1-1=-2

x₂=-1-1/2=-1.5

-1.5-(-2)=0.5

because of x₂>x₁ the sequence monotonously increases

Since for all n,xn>0 $\lim _{n\to \infty }\left(-1-\frac{1}{n}\right)=-1$ then it is bounded above

which means that the limit of the set exists.