I am confused when I have tried to prove the given inequality for Semi-Regular Continued Fractions below
$$ \frac{1}{a_1+1}\leq \frac{1}{a_1+\frac{\epsilon_2}{a_2+\frac{\epsilon_3}{a_3+\frac{\epsilon_4}{\cdots \frac{\epsilon_n}{a_n}}}}}\leq 1 $$ or $$ \frac{1}{a_1+\frac{\epsilon_2}{a_2+\frac{\epsilon_3}{a_3+\frac{\epsilon_4}{\cdots \frac{\epsilon_n}{a_n}}}}} \in [\frac{1}{a_1+1}, 1] $$
Let me give the definition of continued fractions.
Definiton. A continued fraction is understood to be a pair of two finite or infinte sequences $(\epsilon_n)_{n\geq1}$, with both $\epsilon_n\in \{\pm1\}, n\geq1$, and $(a_n)_{n\geq0}$, with $a_n\in \mathbf{Z}, n\geq0$, where in the finite case both sequences and with the same index. In the finite case we denote by $$ a_0+\frac{\epsilon_1}{a_1+\frac{\epsilon_2}{a_2+\frac{\epsilon_3}{a_3+\frac{\epsilon_4}{\cdots \frac{\epsilon_n}{a_n}}}}} $$
Definiton. A finite or infinite continued fraction is called Semi-Regular when $a_0\in\mathbf{Z}$; $a_n\in \mathbf{N}, n\geq1; \epsilon_{n+1}+a_n\geq1, n\geq1,$ and, in the infinite case, $\epsilon_{n+1}+a_n\geq2$ infintely often.
What I am trying to do is to prove the following:
Let
$$\frac{1}{a_1+\frac{\epsilon_2}{a_2+\frac{\epsilon_3}{a_3+\frac{\epsilon_4}{\cdots \frac{\epsilon_n}{a_n+ \cdots}}}}}$$ be a (finite or infinite) Semi-Regular Continued Fraction and let $(\frac{p_n}{q_n})_{n\geq-1}$ be its sequence of convergents. A simple induction arguments shows that
$$ \frac{1}{a_1+\frac{\epsilon_2}{a_2+\frac{\epsilon_3}{a_3+\frac{\epsilon_4}{\cdots \frac{\epsilon_n}{a_n}}}}} \in [\frac{1}{a_1+1}, 1]. $$ But it is unconvenient for me to use induction method in this case.