Triangular Numbers

Let $\mathbb{N}$ be the set of natural numbers, with addition as in that definition and with the order $<$; let $\mathbb{R}$ be the real numbers, an ordered field with multiplicative identity $1$; and let $\iota:\mathbb{N}\to\mathbb{R}$ be the canonical map of $\mathbb{R}$. Write $2=1+1$ in $\mathbb{R}$.

Then the following hold.

1. (Existence and uniqueness.) For every $k\in\mathbb{N}$ there is exactly one natural number $T_{k}$ with

$$2\,\iota(T_{k})=\iota(k)\bigl(\iota(k)+1\bigr).$$

The family $(T_{k})_{k\in\mathbb{N}}$ so determined is a sequence in $\mathbb{N}$; its terms are called the triangular numbers.

2. (Base and step.) $T_{1}=1$, and $T_{k+1}=T_{k}+k+1$ for every $k\in\mathbb{N}$.

3. (Strict monotonicity.) If $k,l\in\mathbb{N}$ and $k<l$, then $T_{k}<T_{l}$.