Lower Bound for Subset

Theorem
Let $$\left({S, \preceq}\right)$$ be a poset.

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 $$T$$.

Proof
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.

Also see

 * Upper Bound for Subset