Suppose $S', S ⊆ \Bbb R$ are two sets such that $S' ⊆ S$. Explain why $\inf(S') ≥ \inf(S)$ and $\sup(S') ≤ \sup(S)$.
What does $S'$ mean?
2025-01-12 19:12:42.1736709162
Inf and Sup question
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Hint: Here $S$ and $S'$ are just two subsets of $\Bbb R$. You could have called them $A$ and $B$, or $X$ and $Y$, or whatever.
Bear in mind that $S \subseteq S'$, and every lower (upper) bound for $S'$ is a lower (upper) bound for $S$. Try to translate that into $\inf$ and $\sup$.