If S is a metric space, then it is well known that C(S), the collection of continuous functions with the metric   is a metric space. Show that if S is compact, then C(S) is separable. Suppose X n...

If S is a metric space, then it is well known that C(S), the collection of continuous functions with the metric

is a metric space. Show that if S is compact, then C(S) is separable.

Suppose X_n
converges weakly to X and the random variables Z_n
are such that

converges to 0 in probability. Prove that Z_n
converges weakly to X . This is known as Slutsky’s theorem.

Suppose X_n
take values in a normed linear space and converge weakly to X. Suppose c_n
are scalars converging to c. Show c_nX_n
converges weakly to c_X
.