Prove that if A and B are sequentially compact in R^n, then the set {(x,y) | x ∈ A, y ∈ B} is a sequentially compact set in R^2n.?

Given S = {(x, y) | x ∈ A, y ∈ B}:

Let {(x(n), y(n))} be a sequence in S.

Since A is sequentially compact in R^n, there exists a subsequence {x(n)_k} which converges to some limit L in R^n. Similarly, since B is sequentially compact in R^n, there exists a subsequence {y(n)_k} which converges to some limit M in R^n.

Hence, {(x(n)_k, y(n)_k)} is a convergent subsequence of {(x(n), y(n))} (which converges to (L, M) in R^(2n)). Hence, S is a sequentially compact set in R^(2n).

I hope this helps!