A simple condition for generalized continued fractions to converge to an irrational number

Let M be a real number greater than 1. We consider continued fractions [0,c1,c2,& mldr;]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,c_{1},c_{2},\ldots ]$$\end{document}, where ci\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{i}$$\end{document} are rational numbers greater than or equal to M (denoted by ci is an element of Q >= M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{i}\in \mathbb {Q}_{\ge M}$$\end{document}) for any positive integer i. In this paper, we give a sufficient condition for such a continued fraction to converge to an irrational number. Specifically, we determine how many non-integers must exist in c1,c2,& mldr;,ci\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{1},c_{2},\ldots ,c_{i}$$\end{document} in order for [0,c1,c2,& mldr;]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,c_{1},c_{2},\ldots ]$$\end{document} to be irrational.