True or False Questions for Real Analysis

True or False, if false give a counter example.

If S is a set of complex numbers and C is a positive real number such that |s| < C for all s in S then there exists a smallest circle in the complex plane which contains all the elements in S.

Every convergent series in a Metric Space is Cauchy.

Every Cauchy series in a Metric space is convergent.

If a series passes the root test, it converges absolutely

If (a_n) is a sequence of positive reals which tends to 0, then \sum_{0, infinity} ((-1)^n)a_n converges

A subspace of a metric space is compact if it is closed and bounded

Suppose f:X->Y is a continuous map of a metric space, and X is compact. Then if U is a closed subset of X, f(U) is a closed subset of Y.

If f and g are two real valued functions on R and both g and fg are continuous, then f is continuous.

Thanks in advance for your help, I'll try to change the list as I answer them.