*6. Let (12d) be a metric space, and ﬁx a point a E Y. For each point p E Y, deﬁne a

function fp : Y —gt; R by Mm) = arm) — den, 00 a) Prove that fp E 0.50). b) Show that fit, — fq||oo = d(p, q) for all p,q E Y (thus, the map (I) :p —gt; fp is an

isometry of Y into Cb(Y)). c) Prove that 050’) is complete in the uniform metric. Let Z be the closure of (NY)

in 05,07). Conclude that Z is complete. Thus, every metric space Y is isometric to a dense subset of a complete metric space

Z. The space Z is called the completion of Y. (It can be shown that Z is unique up

to isometry, but you do not need to prove this.)

Math

Please help me prove the following concerning sequence of functions where C_b(Y) is the space of all continuous