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

*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.)
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