Answer to Question #170518 in Real Analysis for Prathibha Rose

Question

Let D is a subset of R² and (x₀,y₀) element of R² be such that D contains S_r(x₀,y₀) \{(x₀,y₀)} for some r>0 and let f:D to R be any function .prove that f(x,y) tends to infinity as (x,y) converges to (x₀,y₀) if and only if the (alpha -delta) condition is true

Answer 1

 The generalization to R

n is that if X1, . . . , Xn are closed subsets of R, then X1 × · · · × Xn

is a closed subset of R

n. We prove this generalized statement, which in particular proves the case

n = 2.

Let (x1, . . . , xn) be a limit point of X1 × · · · × Xn. So there exists a sequence (x

(k)

1, . . . , x(k)n ) in

X1 ×· · ·×Xn which converges to (x1, . . . , xn). By Theorem 37.2, we have limk→∞ x

(k)

j = xj for each

j = 1, . . . , n. Hence xj is a limit point of Xj for each j. Since each Xj is closed, we have xj ∈ Xj

for each j. Hence (x1, . . . , xn) ∈ X1 × · · · × Xn