subject: advance calculus topic: the set of real numbers

Hello tutors I just need a help to explain this to me because I dont know how my groupmate solve number 10 question. Does it have reference? or how did he solve it

here is the question

P (10) Define the metric d on R? by d( (x1, y1). (x2, y2) ) = max{ |x - x2l. ly1 - y2| ). Verify that this is a metric on R' and for & > 0, draw an arbitrary -neighborhood for a point (x, y) e R2. Success never rests. On your worst days, be good; On your best days, be great; and on every other day, be better. - Carmen MarianoWe are given a metric space in R2 d(12 , , y . ) , ( 22 , 42 ) ) = max$ 121 - x21 , ly-yzly To proved is mebic space O d( a, y ) zo as la - y / zo max $ 12 - 913 2 0 > d ( ( 2 1 , y .) , (2 2 , yz ) ) = max$121 - 221, ly1- yzly 20 d ( 2 ,4 ) = 0 d ( ( 2 1 , y, ) , ( 2 2 , 42 ) ) 20 ") max$ 121 - 221 , ly1- 9213 1) 121- 221 =0 and ly,-421 1) 21 = 12 and yizyz ( x, , y 1 ) = ( 2 2 , yz ) 3 d( (xi, yi ) , ( 212 , yz) ) ~ { max $ 1x1 - 721 , ly,- y213 } 2 max { laz - 211 , 142 - yily z d [ = ( 2 , yz ) , ( 21 , y.) ).d( (21, yi), ( 23 , 43) ) 2 max( 121 - x31 , ly1 - 4313 2 maxe la, - 2 2 + 72 - 23 1 , 1 91 - yz + yzthy 3) 13 2 max$ 1 ( 2 1 - 2 2 ) + ( 2 2 - 23 ) 1 , 1( y , - yz). + (yz -y 3)13 [ max & la1 - 221, ly1 - 4213 + max8 122-231, ly2-431} s d ( ( 2 ( 1 , y , ) , ( 2 4, y z ) ) + d ( ( x 2 , yz ) , ( 23 , y3) ) Thes , the given function is a mebic space . ( x , y ) ( 3 - R ) (7-E ) (ate ) The neighbourhood will be of the form of a rectangle here as E is same for a andy it will bea square with midpoint ( a, y )