7. Suppose that E is a compact subset of a metric space X.

(a) If I,g : E -+ Rn are uniformly continuous, prove that 1+ 9 and I· 9 are uniformly continuous. Did you need compactness for both results?

(b) If g: E -+ R is continuous on E and g(x) ¥- 0 for x E E, prove that l/g is a bounded function.

(c) If I,g: E -+ R are uniformly continuous on E and g(x) ¥- 0 for x E E, prove that 1/ 9 is uniformly continuous on E.