Consider two metrics on R. Let d(x,y):=∣x−y∣ be the standard metric, and let d′(x,y):=1+∣x∣x−1+∣y∣y.
a) Show that (R,d′) is a metric space (if you have done Exercise 7.3.10, the computation is the same).
b) Show that the topology is the same, that is, a set is open in (R,d) if and only if it is open in (R,d′).
c) Show that a set is compact in (R,d) if and only if it is compact in (R,d′).
d) Show that a sequence converges in (R,d) if and only if it converges in (R,d′).
e) Find a sequence of real numbers that is Cauchy in (R,d′) but not Cauchy in (R,d).
f) While (R,d) is complete, show that (R,d′) is not complete.