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 c₁,c₂,…, cᵢ in order for the continued fraction [0; c₁,c₂,…] to be irrational, where cᵢ are rational numbers greater than $M>1$ (denoted by cᵢ∈ Q≥ M) and the denominator of cᵢ 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;c₁,c₂,…], where cᵢ 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.