Let (n)-1 be a sequence in R, and let x be a vector in R. Consider...

 

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Let (n)-1 be a sequence in R, and let x be a vector in R. Consider the following four statements: (1) (n)-1 converges to x with respect to the taxi-cab distance 7 (associated to ||·||1). (2) (n)-1 converges to x with respect to the Euclidean distance d (associated to ||·||2). (3) (n)-1 converges to x with respect to the max-distance p (associated to ||· ||∞). (4) (n)-1 converges to x componentwise (in the sense of the preceding definition). please (a) Prove the implication (3)⇒(4), and show by example that the converse of this implication is not true. (b) Prove the implication (1)⇒(2). (c) Show by example that the converse of the implication (2)⇒(3) is not true. Let (n)-1 be a sequence in R, and let x be a vector in R. Consider the following four statements: (1) (n)-1 converges to x with respect to the taxi-cab distance 7 (associated to ||·||1). (2) (n)-1 converges to x with respect to the Euclidean distance d (associated to ||·||2). (3) (n)-1 converges to x with respect to the max-distance p (associated to ||· ||∞). (4) (n)-1 converges to x componentwise (in the sense of the preceding definition). please (a) Prove the implication (3)⇒(4), and show by example that the converse of this implication is not true. (b) Prove the implication (1)⇒(2). (c) Show by example that the converse of the implication (2)⇒(3) is not true.