How to understand “有理数贯逼近实数”？
"有理数" = rational numbers
"数贯" = (numerical) sequence (obsolete, the more conventional term nowadays is "数列")
"逼近" = literally, "to approach", "to get close to"； mathematically, "to approximate"
"实数" = real numbers

Therefore, "有理数贯逼近实数" means "using a sequence of rationals to approximate (a) real", so this sounds like something similar to a Cauchy sequence.

In context of this article, it is probably that Hua Loo-Keng was giving a lecture on the construction of real numbers using Cauchy sequences (likely in a real analysis course), and he introduced continued fractions (连分数) in the context of that.

Disclaimer: I am a native Chinese (Mandarin) speaker with a reasonable amount of knowledge of mathematics (but I am not a professional mathematician).