Closed Form for Triangular Numbers/Proof using Cardinality of Set

Proof
Let $\N_n^* = \set {1, 2, 3, \cdots, n}$ be the initial segment of natural numbers.

Let $A = \set {\tuple {a, b}: a \le b, a, b \in \N_n^*}$

Let $B = \set {\tuple {a, b}: a \ge b, a, b, \in \N_n^*}$

Let $\phi: A \to B$ be the mapping:
 * $\map \phi {x, y} = \tuple {y, x}$

By definition of dual ordering, $\phi$ is a bijection:
 * $(1): \quad \size A = \size B$

We have:

Thus:

Combined with $\left({1}\right)$ this yields:
 * $\size A = \dfrac {n^2 + n} 2 = \dfrac {n \paren {n + 1} } 2$

It remains to prove that:
 * $T_n = \size A$