MAT 125a, HW4 Solutions 10 Suppose that limn pn = p in a given metric space. The set of points S = cfw_p, p1 , p2 . is closed. lemma: If x cfw_a1 , a2 , a3 , . and for all > 0 B (x) cfw_a1 , a2 , . = then > 0 and n N we have B (x) cfw_an+1 , an+2 , an+3 .
MAT 125a, HW3 Solutions 1(a) Consider the set Rn of n-tuples of real numbers with
n
d(x, y) =
i=1
|xi yi |.
We wish to verify that d denes a metric on Rn : (1) Since d(x, y) is a sum of non-negative numbers we know that d(x, y) 0 for all x, y Rn . (2) If
MAT 125a, HW2 Solutions 7 (a) (maxcfw_a, b =
a+b+|ab| ): 2
There are two cases: a b and a < b.
If a b then we have a + b + |a b| a+b+ab 2a = = = a = maxcfw_a, b 2 2 2 . If a < b then we have a + b ( a b) 2b a + b + |a b| = = = b = maxcfw_a, b 2 2 2 . In e