The set $S = \{1/n:n\in\Bbb{N}\}$ is not open, because for example it contains the point $1$ but it does not contain any $\varepsilon$-neighbourhood...
See my comment on your previous thread. The way this proof works in that you divide the real line into subintervals of length $1/2^n$ and you locate...
No. What is happening in this proof is that you narrow down the location of $\sup(S)$ by an approximation process. You start from a point $s$ in $S$,...
Let $t = \sup(-S)$. You want to show that $-t = \inf(S)$. To do that, you could certainly use Prop. 2.1.30. But you should also be able to do it from...
I am interpreting the question to mean that we are looking for the largest value of $K$ such that $(a+b+c+d)^2 \geqslant K b c$ for all real numbers...