Floor of Simple Finite Continued Fraction

Theorem
Let $(a_k)_{k\geq0}$ be a simple finite continued fraction of length $n \geq 0$.

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

Then the floor of $x$ is the partial denominator $a_0$:
 * $\lfloor x \rfloor = a_0$.

unless $n = 1$ and $a_1 = 1$, in which case $x = \lfloor x \rfloor = a_0 + 1$.

Length 0
If $n=0$, we have $x = [a_0, \ldots, a_n] = a_0$ by definition of value.

By Floor Function of Integer, $\lfloor x \rfloor = \lfloor a_0 \rfloor = a_0$.

Length 1
If $n = 1$, then $x = [a_0, a_1] = a_0 + \dfrac 1{a_1}$.

If $a_1 > 1$, then $a_0 < a_0 + \dfrac 1{a_1} < a_0 + 1$.

By definition of floor function, $\lfloor x \rfloor = a_0$.

If $a_1 = 1$, then $x = a_0 + 1$.

By Floor Function of Integer, $\lfloor x \rfloor = a_0 + 1$.

Length at least 2
By Value of Finite Continued Fraction of Real Numbers is at Least First Term:
 * $[a_0, \ldots, a_n] > a_0$
 * $[a_1, \ldots, a_n] > a_1 \geq 1$

Thus:
 * $a_0 < [a_0, \ldots, a_n] = a_0 + \dfrac 1 {[a_1, \ldots, a_n]} < a_0 + 1$.

By definition of floor function, $\lfloor x \rfloor = a_0$.

Also see

 * Properties of Value of Finite Continued Fraction