Therefore, we can see that B ε, ρ p ( x ) ⊂ B ε, ρ inf ( x ) ⊂ B ε, n 1 ∕ p ρ p ( x ) Which means that a subset is open in L p metric if and only if it is open in L ∞ metric.
Pick x, y ∈ ℝ n. We can see that max i | x i − y i | ≤ ( ∑ i | x i − y i | p ) 1 ∕ p ≤ n 1 ∕ p max i | x i − y i | Since: | x j − y j | ≤ max i | x i − y i | And: ∑ i | x i − y i | p ≤ n p max i | x i − y i |.