Continued Fraction Expansion of Euler's Number/Proof 1

Proof
From the recursive definition of continued fractions, we have:

Let:

In other words:
 * $a_{3 n + 1} = 2 n$

and:
 * $a_{3 n + 0} = a_{3 n + 2} = 1$

Then $p_i$ and $q_i$ are as follows:


 * $\begin{array}{r|cccccccccc}

\displaystyle i & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9  \\ \hline

p_i & 1 & 1 & 2 & 3 & 8 & 11 & 19 & 87 & 106 & 193 \\

q_i & 1 & 0 & 1 & 1 & 3 & 4 & 7 & 32 & 39 & 71 \\ \hline

\end{array}$

Furthermore, $p_i$ and $q_i$ satisfy the following $6$ recurrence relations:

Our ultimate aim is to prove that:


 * $\displaystyle \lim_{n \mathop \to \infty} \frac {p_n} {q_n} = e$

In the pursuit of that aim, let us define the integrals:

Lemma
We assert that $A_n$, $B_n$ and $C_n$ all converge to $0$ as $n \to \infty$:

We now have:

from which we conclude: