MA2223: SOLUTIONS TO PROBLEM SHEET 1
1. Show that the diameter of an open ball B(a, r) in Euclidean space
Rn is exactly 2r.
Solution: We proved in class that in any metric space the diameter of an open ball is at most twice the radius. So we know
diam(B(a, r)) ≤ 2r. Let’s suppose diam(B(a, r)) < 2r and try
to obtain a contradiction. We will need to work with coordinates
so write a = (a1 , . . . , an ). Let t = diam(B(a, r)) and consider the
following two points,
t r
x = (a1 + + , a2 , . . . , an )
4 2
t
y = (a1 − , a2 , . . . , an )
2
We claim that x and y are contained in B(a, r) and that d(x, y) >
diam(B(a, r)). This would of course be a contradiction. We have
r r
t r
d(a, x) = + < + = r
4 2
2 2
t
d(a, y) = < r
2
which shows x and y are contained in B(a, r). Now we observe that
t
t
t
t
t r
d(x, y) = + + > + + = t
4 2 2
4 4 2
This gives us our contradiction and so we conclude that diam(B(a, r))
must be exactly 2r.
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