# EE6415: Nonlinear Systems Analysis Tutorial 4 solution

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1. Let X be the set of all ordered triples of zeros and ones. Show that X consists of eight elements
and a metric d on X is defined by d(x, y) = number of places where x and y have different entries.
For example, d(010, 111) = 2. (This metric is called Hamming distance).

2. Show that the function d on the set X defined by
d(x, y) = Z b
a
|x(t) − y(t)|dt
is a metric, where X is the set of all real-valued functions x, y, · · · which are functions of an
independent real variable t and are defined and continuous on a given closed interval J = [a, b].

3. Consider the space of all sequences x = (ζi), y = (ηi). Prove that
d(x, y) = X∞
j=1
1
2
j
|ζj − ηj |
1 + |ζj − ηj |
is a metric. Further, show that
d2(x, y) = X∞
j=1
rj
|ζj − ηj |
1 + |ζj − ηj |
is a metric for any sequence (rj ) for which every element is positive, and Prj converges.

4. Show that d(x, y) = p
|x − y| is a metric on the set of Real Numbers.

5. Show that a Cauchy sequence is bounded. Is boundedness of a sequence in a metric space sufficient
for the sequence to be Cauchy? Convergent?

6. Let d be a metric on X. Determine all constants k such that
(a) kd
1
(b) d + k
is a metric on X

7. Does d(x, y) = (x − y)
2 define a metric on the set of all real numbers?

8. The triangle inequality has several useful consequences. Show that the following inequalities are
true for any metric d
(a) |d(x, y) − d(z, w)| ≤ d(x, z) + d(y, w)
(b) |d(x, z) − d(y, z)| ≤ d(x, y)

9. Consider the normed linear vector space of rational numbers Q with norm ∥x∥ = |x|. For each of
the sequences an given next, find whether an is (a) convergent in Q (b) a Cauchy sequence.
(a) an = 1 + 1
2 + · · · +
1
n
(b) an =
Fn+1
Fn
where Fn is the n
th Fibonacci Number.