how to add supremums

Question

I need to prove that $$\sup(S)+\sup(T)=\sup(S+T)$$ I don't understand what $\sup(S+T)$ means, can you show me examples for sets $S$ and $T$ for which this equation holds?

Answer 1

For example, take $S=\{1,\frac12,\frac13,\frac14,\dots\} = \{ \tfrac1m\colon m\ge1\}$ and $T=\{0,\frac12,\frac23\frac34,\dots\} = \{ 1-\tfrac1n\colon n\ge1\}$. So $\sup S=\sup T=1$ in this case, and $$ S+T = \{s+t\colon s\in S,\,t\in T\} = \{ \tfrac1m + (1-\tfrac1n)\colon m,n\ge1\} = \{ 1+\tfrac1m -\tfrac1n \colon m,n\ge1\}. $$ You can check that $\sup(S+T)=2$ in this case (as your equation says it should): every element of $S+T$ is less than $2$, while the subset $\{1\}+T=\{ 2-\tfrac1n\colon n\ge1\}$ contains elements arbitrarily close to $2$.