Non-Empty Subset of Initial Segment of Natural Numbers has Greatest Element

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Theorem

Let $n \in \N_{>0}$ be a non-zero natural number.

Let $\N^*_n$ denote the Initial segment $\set {1, 2, \ldots, n}$ of the non-zero natural numbers.


Then every non-empty subset of $\N^*_n$ has a greatest element.


Proof

The proof will proceed by the Principle of Mathematical Induction on $\N_{>0}$.

Let $S$ be the set defined as:

$S := \left\{ {n \in \N:}\right.$ Every non-empty subset of $\N^*_n$ has a greatest element$\left.{}\right\}$


Basis for the Induction

We have that:

$\N^*_1 = \set 1$

The only non-empty subset of $\set 1$ is $\set 1$ itself.

This has a greatest element $1$.


So $1 \in S$.

This is our basis for the induction.


Induction Hypothesis

It is to be shown that, if $k \in S$ where $k \ge 1$, then it follows that $k + 1 \in S$.

This is the induction hypothesis:

Every non-empty subset of $\N^*_k$ has a greatest element.


It is to be demonstrated that it follows that:

Every non-empty subset of $\N^*_{k + 1}$ has a greatest element.


Induction Step

This is our induction step:

Let $T \subseteq \N^*_{k + 1}$.

Either $k + 1 \in T$ or $k + 1 \notin T$.

If $k + 1 \notin T$ then $T \subseteq \N^*_k$.

In that case, then from the induction hypothesis:

$T$ has a greatest element.

If $k + 1 \in T$ then:

$\forall x \in T: x \le k + 1$

and so $k + 1$ is the greatest element of $T$.

So in both cases $T$ has a greatest element and so $k + 1 \in S$.


So $k \in S \implies k + 1 \in S$ and the result follows by the Principle of Mathematical Induction:

$\forall n \in \N$: every non-empty subset of $\N^*_n$ has a greatest element.

$\blacksquare$


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