Lower Bound for Subset

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Let $\left({S, \preceq}\right)$ be an ordered set.

Let $L$ be a lower bound for $S$.

Let $\left({T, \preceq}\right)$ be a subset of $\left({S, \preceq}\right)$.

Then $L$ is a lower bound for $T$.


By definition of lower bound:

$\forall x \in S: L \preceq x$

But as $\forall y \in T: y \in S$ by definition of subset, it follows that:

$\forall y \in T: L \preceq y$.

Hence the result, again by definition of lower bound.


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