The set B is given by
B = { n/n+1 | n = a natural number } as a subset of the Real numbers. Is this set
i) Open
ii) Closed
Provide justification.
The set is not open. Note, $\displaystyle 1/2 \in B$ but if we choose $\displaystyle \epsilon$ small enough then $\displaystyle (1/2-\epsilon,1/2+\epsilon)$ does not lie in $\displaystyle B$, take for example $\displaystyle \epsilon = .1$.
The set is not closed. We need to show that its complement is not open. Let $\displaystyle B^*$ be its complement. Since $\displaystyle 1\not \in B$ it means $\displaystyle 1\in B^*$ but not interval $\displaystyle (1-\epsilon,1+\epsilon)$ lies completely outside $\displaystyle B$ because $\displaystyle n/(n+1)$ converges to $\displaystyle 1$.