Dear reader, don't laugh: I had thought I already understood subsequences, but then it turned out that I was mistaken. I should have noticed the vague, unverbalized discomfort I felt about the subscripted-subscript notation, \((a_{n_k})\). But really it shouldn't be confusing at all: as Bernd S. W. Schröder points out in his Mathematical Analysis: A Concise Introduction, it's just a function composition. If it helps (it helped me), say that \((a_n)\) is mere syntactic sugar for \(a(n): \mathbb{N} \to \mathbb{R}\), a function from the naturals to the reals. And \((a_{n_k})\) is just the composition \(a(n(k))\), with \(n(k): \mathbb{N} \to \mathbb{N}\) being a strictly increasing function from the naturals to the naturals.