Question based on infimum concept Somebody told me that infimum of smaller set is always bigger than infimum of bigger set .I couldnot understand it how ? How can it be justified ?--prophetes.ai

Question based on infimum concept Somebody told me that infimum of smaller set is always bigger than infimum of bigger set .I couldnot understand it how ? How can it be justified ?

Not "always bigger" -- "never smaller." To see why, take the infimum $m$ of the bigger set $S$. For all $s\in S$, $m \leq s$ by definition. But for any subset $T\subseteq S$, you have that for any $t\in T$, $t\in S$: so $m \leq t$ as well. That is, $m$ is a lower bound on $T$ -- in particular, it is not greater than the infimum of $T$, which is the _greatest_ lower bound for $T$ possible.