Value of Finite Continued Fraction of Real Numbers is at Least First Term

Theorem
Let $(a_0, \ldots, a_n)$ be a finite continued fraction in $\R$ of length $n \geq 0$.

Let the partial quotients $a_k>0$ be strictly positive for $k>0$.

Let $x = [a_0, a_1, \ldots, a_n]$ be its value.

Then $x \geq a_0$, and $x>a_0$ if the length $n\geq 1$.

Proof
If $n=0$, we have $x = [a_0] = a_0$ by definition of value.

Let $n>0$.

By definition of value:
 * $[a_0, a_1, \ldots, a_n] = a_0 + \dfrac 1 {[a_1, a_2, \ldots, a_n]}$

By Value of Finite Continued Fraction of Strictly Positive Real Numbers is Strictly Positive:
 * $[a_1, a_2, \ldots, a_n] > 0$.

Thus
 * $[a_0, a_1, \ldots, a_n] = a_0 + \dfrac 1 {[a_1, a_2, \ldots, a_n]} > a_0$

Also see

 * Properties of Value of Finite Continued Fraction