Set Metric

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real analysis/ prove closed set / metric spaces?

Prove that {(a1, a2) in R^2 : 0 <= a1 <= 2, 0 <= a2 <= 4} is a closed set in the Euclidean metric.

Set A ⊂ ℝ² to be the set specified above.
We need to show that every element of ℝ² A is an interior point.

Let x = (x₁,x₂) ∈ ℝ² A and set r = min{|x₁|, |x₁ - 2|, |x₂|, |x₂ - 4|} (x ∉ A so r > 0);
so for every a = (a₁,a₂) ∈ A, |x - a| = √((x₁ - a₁)² + (x₂ - a₂)²) ≥ |x₁ - a₁|,|x₂ - a₂| ≥ r.

Let p ∈ B(x,r) (the open ball of radius r at x). By the triangle inequality we have for every a ∈ A that |p - a| ≥ |x - a| - |x - p| > 0.
Hence B(x,r) ⊂ ℝ² A.

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