The topologies on the infinite-dimensional space ℓ ℓ p are inequivalent for different p . For example, the sequence of vectors x n = ( 2 − − n , 2 − − n , … … , 2 − − n , 0 , 0 , … … ) , in which the first 2 n components are 2 − − n and the following ones are 0 , converges to the zero vector for p = ∞ ∞ , but does not for p = 1 : ‖ ‖ x n ‖ ‖ ∞ ∞ = sup ( 2 − − n , 0 ) = 2 − − n → → 0 , but ‖ ‖ x n ‖ ‖ 1 = ∑ ∑ i = 1 2 n 2 − − n = 2 n ⋅ ⋅ 2 − − n = 1.