Let $[a_0,a_1,a_2,\ldots]$ be an infinite simple continued fraction.
How can one show that $a_0$ is the integral part of $[a_0,a_1,a_2,\ldots]$?
There is a similar theorem for finite continued fractions with length $n\geq 1$ and $a_n \geq 2$ that I know, however I don't know how to extend this to the infinite case. The infinite continued fraction is the limit of the sequence of its convergents but as each convergent will not always have $2$ as its last partial coefficient, I don't know how to proceed.