Let Rn be endowed with the usual metric and suppose that {xk} is a sequence in R

Question:

Let Rn be endowed with the usual metric and suppose that {xk} is a sequence in R" with components xk(j); that is,

a) Use Remark 8.7 to prove that {xk} is bounded in Rn if and only if there is a C > 0 such that |xk(j)| b) Let a ˆˆ Rn. Prove that xk: †’ a as n †’ ˆž if and only if xk(j) †’ ak, as k †’ ˆž, for every j ˆˆ {1, 2,..., n}.
c) Find the limit of each of the following sequences.