Finite Set Contains Subset of Smaller Cardinality

Theorem
Let $S$ be a finite sets.

Let
 * $\size S = n$

where $\size {\, \cdot \,}$ denotes cardinality.

Let $0 \le m \le n$.

Then there exists a subset $X \subseteq S$ such that:
 * $\size X = m$

Case 1
Let $m = n$.

Then $X = S$ is a subset $X \subseteq S$ such that:
 * $\size X = m$

Case 2
Let $m = 0$.

Then $X = \O$ is a subset $X \subseteq S$ such that:
 * $\size X = m$

Case 3
Let $0 < m < n$.

By definition of the cardinality of a finite set:
 * $S \sim \N_{< n}$

where:
 * $\sim$ denotes set equivalence
 * $\N_{<n}$ is the set of all natural numbers less than $n$

Consider the set of all natural numbers less than $m$, $\N_{< m}$.

By the definition of a subset:
 * $\N_{< m} \subseteq \N_{< n}$

Let $i : \N_{< m} \to \N_{< n}$ be the inclusion mapping.

Let $f : \N_{< n} \to S$ be a bijection between $\N_{< n}$ and $S$.

From Composite of Injections is Injection, the composite $f \circ i : \N_{< m} \to S$ is an injection.

Let $X = \Img {f \circ i}$ be the image of $f \circ i$.

By definition of the image of a mapping:
 * $X \subseteq S$

From Injection to Image is Bijection:
 * $\N_{< m} \sim X$

By definition of the cardinality of $X$:
 * $\size X = m$

Also see

 * Cardinality of Subset of Finite Set