# Square number

## Definition 1

\begin{align} S_0 &= 0 \\ S_{n + 1} &= S_n + 2(n + 1) - 1 \end{align}

## Definition 2

$$S_n = \sum_{i = 0}^{n} 2i - 1$$

## Definition 3

$$S_n = n^2$$
LEMMA

Definitions one and two are equivalent.

Proof available

LEMMA

Definitions one and three are equivalent.

Proof available

THEOREM

All three definitions are equivalent.

Proof unavailable

THEOREM    The square numbers are strictly increasing.

Proof unavailable