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Let (X, dX) and (Y, dY ) be metric spaces, and let f : X → Y be continuous. Define the distance d on the product space X × Y as in class, so that (X × Y, d) is a metric space. Show that the graph Γf of f, defined by Γf = {(x, y) ∈ X × Y | f(x) = y}, is a closed subset of X × Y .

Exercise 4 ( 6 points ) . Let ( * 1 , di ) . …. ( * n , An ) be metric spaces and let * _ _ Xi. EquipI with a metric the way seen in class . Let ( ack ) & C X. For all KE N we can write ack( OCK, 1 , *Ck, 2 1 … . 2 kin ) , where for each it [ ] . …. ~’ the number ack.; E X; is the ith coordinate of ack`Let a = ( a1 , 02 . … . an ) EX.Prove that limk ack = a if and only if limk ack,; – di for all it , …. ~ ].