Explore the concept of infinite sequences in this 37-minute lecture from the Math Foundations series. Delve into the distinction between finite sequences and "ongoing sequences" or "on-sequences," focusing on three key examples: natural numbers, prime numbers, and Catalan numbers. Examine Euclid's proof of prime number infinity and discuss potential computational challenges. Investigate various definitions and formulas for the Catalan sequence, highlighting the indirect nature of specifying on-sequences. Consider the linguistic relationship between "finite" and "infinite" and its practical implications. Learn new terminology and gain insights into the fundamental differences between finite sequences and on-sequences in mathematics.