Infimum of Subset of Bounded Below Set of Real Numbers

Theorem
Let $A$ and $B$ be sets of real numbers such that $A \subseteq B$.

Let $B$ be bounded below.

Then:
 * $\inf A \ge \inf B$

where $\inf$ denotes the infimum.

Proof
Let $B$ be bounded below.

By the Continuum Property, $B$ admits an infimum.

By Subset of Bounded Below Set is Bounded Below, $A$ is also bounded below.

Hence also by the Continuum Property, $A$ also admits a infimum.

$\inf A < \inf B$.

Then:
 * $\exists y \in A: y < \inf B$

Thus by definition of infimum, $y \notin B$.

That is:
 * $A \not \subseteq B$

which contradicts our initial assumption that $A \subseteq B$.

Hence the result by Proof by Contradiction.