Let ||.||_1 and ||.||_2 be norms on a vector space X

such that X_1 = (X, ||.||_1) and X_2=(X,||.||_2) are complete.

If ||x_n||_1 ---> 0 always implies ||x_n||_2 -----> 0,

show that convergence in X_1 implies convergence in X_2.

and conversely show that there are positive numbers

a and b such that for all x element of X,

a||x||_1 <= ||x||_2 <= b ||x||_1.