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

Theorem

Let $\sequence {a_0, \ldots, a_n}$ be a finite continued fraction in $\R$ of length $n \ge 0$.

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

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

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

Proof

If $n = 0$, we have $x = \sqbrk {a_0} = a_0$ by definition of value.

Let $n>0$.

By definition of value:

$\sqbrk {a_0, a_1, \ldots, a_n} = a_0 + \dfrac 1 {\sqbrk {a_1, a_2, \ldots, a_n} }$

By Value of Finite Continued Fraction of Strictly Positive Real Numbers is Strictly Positive:

$\sqbrk {a_1, a_2, \ldots, a_n} > 0$

Thus:

$\sqbrk {a_0, a_1, \ldots, a_n} = a_0 + \dfrac 1 {\sqbrk {a_1, a_2, \ldots, a_n} } > a_0$

$\blacksquare$

Also see

• Properties of Value of Finite Continued Fraction