# Understanding Proof about Continued Fraction convergent sequences

I copied a proof from lecture and don't understand the end of it. It is intro number theory on continued fractions. Hopefully someone can explain it to me

Background: The sequences {$$h_n$$} and {$$k_n$$} are defined recursively as such.

$$h_{-2} = 0 \quad h_{-1} = 1 \quad h_i = a_ih_{-1} + h_{-2}$$

$$k_{-2} = 1 \quad k_{-1} = 0 \quad k_i = a_ik_{-1} + k_{-2}$$

Prop:

$$det\begin{vmatrix} h_i & h_{i-1} \\ k_i & k_{i-1} \\ \end{vmatrix} = (-1)^{i-1}$$

Proof:

Works for $$i=0$$ (I'm omitting this part, I understand we need the base case).

(Then the next part is where I get confused)

Suppose true for $$i < n$$

Then

$$det\begin{vmatrix} h_n & h_{n-1} \\ k_n & k_{n-1} \\ \end{vmatrix} = h_nk_{n-1}-k_nh_{n-1} = (a_nh_{n-1}+h_{n-2})k_{n-1} - (a_nk_{n-1}+k_{n-2})h_{n-1} = h_{n-2}k_{n-1}- k_{n-2}h_{n-1}$$

(Following all the algebra and substitution up to this point.)

Then:

$$= -(h_{n-1}k_{n-2}-k_{n-1}h_{n-2}) = -(-1)^{n-2} = (-1)^{n-1}$$ By induction hypothesis we are done.

Some questions:

1. How are we done? I don't understand, don't we need to do an $$n+1$$ case? If someone could also shed light on induction proofs where the induction hypothesis is of the type $$i that'd really help me out. Maybe I'm just missing something or confusing myself.
• We’re assuming it’s true for $n-1$ and showing it’s true for $n$; by the way, I think you meant $n$ not $i$ in the subscripts in the second determinant – J. W. Tanner Apr 22 at 15:31
• Changes the subscripts in the second det, thank you. I just don't understand why we multiply the last part by $-1$ and how the part inside the brackets $= -(-1)^{n-2}$ – Mathstatsstudent Apr 22 at 15:33
• $h_{n-1}k_{n-2}-k_{n-1}h_{n-2}= (-1)^{n-2}$ is basically the statement (assumed true) for $n-1$ – J. W. Tanner Apr 22 at 15:46

To prove something by induction, we have to prove it for the base case, say $$i=0,$$
and then prove that, if it holds for any $$m ≥ 0$$, it holds for $$m+1.$$
Equivalently (replacing $$m+1$$ with $$n$$), if it holds for any $$n-1\ge0$$, it holds for $$n$$.
So we assume the statement is true for any $$n-1$$, i.e., $$h_{n-1}k_{n-2}-k_{n-1}h_{n-2}= (-1)^{n-2}$$
and then show that the statement holds for $$n$$, i.e., $$h_nk_{n-1}-k_nh_{n-1}=(-1)^{n-1}$$.