Smallest Element is Lower Bound
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Theorem
Let $\struct {S, \preceq}$ be an ordered set.
Let $T \subseteq S$.
Let $T$ have a smallest element $m \in T$.
Then $m$ is a lower bound of $T$.
It follows by definition that $T$ is bounded below.
Proof
Let $m \in T$ be a smallest element of $T$.
By definition:
- $\forall y \in T: m \preceq y$
But as $T \subseteq S$, it follows that $m \in S$.
Hence:
- $\exists m \in S: \forall y \in T: m \preceq y$
Thus $T$ is bounded below by the lower bound $m$.
$\blacksquare$
Also see
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.7$: Maximum and Minimum