Existence of Chebyshev Polynomials of the First Kind

Theorem
There exists a Chebyshev polynomial of the first kind for all natural numbers $n$.

Proof
For $n = 0$:


 * $\map {T_0} x = 1$, $T_0 \in \Bbb P$

For $n = 1$:


 * $\map {T_1} x = x$, $T_1 \in \Bbb P$

Assume $\map {T_n} {\cos \theta} = \map \cos {n \theta}$.


 * $\map {T_{n+1} } x = 2x \map {T_n} x - \map {T_{n - 1} } x$, $T_{n+1} \in \Bbb P$

By the Second Principle of Mathematical Induction, $T_n \in \Bbb P$ for all natural numbers $n$.