Subset of Bounded Below Set is Bounded Below

Theorem

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

Let $B$ be bounded below.

Then $A$ is also bounded below.

Proof

Let $B$ be bounded below.

Then by definition $B$ has a lower bound $L$.

Hence:

$\forall x \in B: x \ge L$

But by definition of subset:

$\forall x \in A: x \in B$

That is:

$\forall x \in A: x \ge L$

Hence, by definition, $A$ is bounded below by $L$.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: Exercise $1.5: 3$