On generalized continued fractions converging to an irrational number

In [5], we give a sufficient condition for generalized continued fractions to converge to an irrational number. Specifically, we determine how many non-integers must exist in $c_{1},c_{2},\ldots, c_{i}$ in order for the continued fraction $[0; c_{1},c_{2},\ldots]$ to be irrational, where $c_{i}$ are rational numbers greater than $M>1$ (denoted by $c_{i}\in \mathbb{Q}_{\ge M}$) and the denominator of $c_{i}$ less than or equal to positive integer $N$. In [6], we extended this result from the case $M>1$ to $M>0$. In this paper, we give a further extension by removing the restrictions on the denominators of the partial quotients. Let $M$ be a real number greater than $0$. We consider continued fractions $[0;c_{1},c_{2},\ldots]$, where $c_{i}$ are rational numbers greater than or equal to $M$ for any positive integer $i$. We give a sufficient condition for it to converge to an irrational number and some interesting examples.