MATH 127C University of Nairobi Cauchy Sequence Mathematics Homework 1

please write the answer and logic clearly.

here is hw and some handout.

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Math 127C – Homework 1
Write complete solutions to each of the following problems. Submit your
answers in PDF format to Gradescope by Monday, June 29 at 11:59 PM.
1. Show that the 1-norm/taxicab norm on Rn , defined by k(x1 , . . . , xn )k1 =
|x1 | + . . . + |xn | for all (x1 , . . . , xn ) ∈ Rn , is a norm.
2. Let V be any nontrivial (V 6= {0}) vector space over R, and let k · k be
a norm on V . Show that for any positive real number r > 0 there exists
x ∈ V such that kxk = r.
3. Let (M, d) be a metric space, and let M be finite. Show that (M, d) is
complete.
4. Let {xn }∞
n=1 be a sequence in a metric space, and suppose the sequence
satisfies the following property: for every  > 0, there exists N ≥ 1 and a
ball B of radius  such that
n > N =⇒ xn ∈ B.
Show that {xn }∞
n=1 is a Cauchy sequence.
5. Show that
S for any family {Uα : α ∈ I} of open sets in a metric space, the
union α∈I Uα is open.
6. Show that if U and V are open sets in a metric space, then U ∩ V is open.
7. Show that a finite subset of a metric space is connected if and only if it
contains exactly one point.
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