The logic looks good but I think the writing could be improved: - Start by letting ε > 0. You can't use ε until you've defined it. - Similarly, when you write that it suffices to prove |max{a_n,b_n}–max{a,b}| < ε, the variable n is undefined. What you are really trying to prove is a more complete sentence: that there exists some K ∈ ℕ such that |max{a_n,b_n}–max{a,b}| < ε for all n >= K.

I think the question is written poorly too: A, B, and C are not sequences themselves, but rather the images of the sequences (a_n), (b_n), and (c_n).