I am a physics student and am self teaching analysis so from time to time would like to get clarification on how accurate my proofs are (no matter how basic they are).
Question: Let $S$ be a nonempty bounded subset of $\mathbb{R}$ .Prove inf$S$ $\leq$ sup$S$.
$Proof:$ Since $S$ is bounded there exists an $s_0$=sup$S$ $\hspace{0.1cm}$ such that $\hspace{0.1cm}$ s$\leq$$s_0$$\hspace{0.1cm}$ and an $\hspace{0.1cm}$ $a_0$=inf$S$ $\hspace{0.1cm}$ such that $a_0 \leq$s $\hspace{0.1cm}$ $\forall s \in S$. Thus $a_0 \leq s \leq s_0$ $\implies$ $a_0 \leq s_0$ $\implies$ inf$S$ $\leq$ sup$S$ []
If people could point out whether I have missed anything obvious or assumed something incorrectly it would be appreciated!