Mistake in Khinchin's "Continued Fractions"

This looks like a bit of a looseness, perhaps in translation. The quoted inequality is not meant to be universally quantified in $l$. If you read the two paragraphs leading up to it, they show that either $q_k < q_{k-1} / (1-a_k)$ or $q_k < q_{k-2} / (1-a_k)$.

Thus the author simply means to say that $(*)$ holds for some $l < k$. If you read the inequality that follows you'll see it is applying this inductively to connect $q_k$ further back to a term earlier than $q_{k_0}$. This repeated application wouldn't be necessary if $(*)$ were true for all $l<k$.