A) provide the sequential criterion for compactness of a set in a metric space.
b) let (mi, di) be metric spaces for i= 1,2. let m=m_{1}\times m_{2} and define the metric d on m by
d(x, y)=d_{1}(x_{1},y_{1})+d_{2}(x_{2},y _{2}),
where x=(x1,x2) and y=(y1,y2). define the projection \pi : m\rightarrow m_{1} by \pi(x)=x_{1}
where x=(x1,x2). explain why direct images of compact sets under \pi are compact.
c) with \pi as defined above, prove that inverse image of compact sets under \pi are also compact provided m2 is compact.