Simple Infinite Continued Fraction is Uniquely Determined by Limit
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Theorem
Let $\sequence {a_n}_{n \mathop \ge 0}$ and $\sequence {b_n}_{n \mathop \ge 0}$ be simple infinite continued fractions in $\R$.
Let $\sequence {a_n}_{n \mathop \ge 0}$ and $\sequence {b_n}_{n \mathop \ge 0}$ have the same limit.
Then they are equal.
Proof 1
Follows immediately from Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself.
$\blacksquare$
Proof 2
Note that by Simple Infinite Continued Fraction Converges, they do indeed have a limit.
The result will be achieved by the Second Principle of Mathematical Induction.
Suppose $\sqbrk {a_0, a_1, a_2, \ldots} = \sqbrk {b_0, b_1, b_2, \ldots}$ have the same value.
First we note that if $\sqbrk {a_0, a_1, a_2, \ldots} = \sqbrk {b_0, b_1, b_2, \ldots}$ then $a_0 = b_0$ since both are equal to the integer part of the common value.
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This is our basis for the induction.
Now suppose that for some $k \ge 1$, we have:
- $a_0 = b_0, a_1 = b_1, \ldots, a_k = b_k$.
Then all need to do is show that $a_{k + 1} = b_{k + 1}$.
Now:
- $\sqbrk {a_0, a_1, a_2, \ldots} = \sqbrk {a_0, a_1, \ldots, a_k, \sqbrk {a_{k + 1}, a_{k + 2}, \ldots} }$
and similarly:
- $\sqbrk {b_0, b_1, b_2, \ldots} = \sqbrk {b_0, b_1, \ldots, b_k, \sqbrk {b_{k + 1}, b_{k + 2}, \ldots} }$.
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As these have the same value and have the same first $k$ partial quotients, it follows that:
- $\sqbrk {a_{k + 1}, a_{k + 2}, \ldots, } = \sqbrk {b_{k + 1}, b_{k + 2}, \ldots}$.
But now $a_{k + 1} = b_{k + 1}$ as each is equal to the integer part of the value of this simple infinite continued fraction.
Hence the result.
$\blacksquare$