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Asked Apr 3, 2019
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pls explain to me step by step. pls dont skip any steps. thanks

metric space is a set M together with a distance function ρ(x,y) that represents the
distance" between elements a and y of the set M. The distance function must satisfy
) plz,y)20 and plr,y) 0 if and only if y:
(ii) p(x,y) o(y,z);
(iii) p(z, y) a(x, z) +a(z, y) for all x, y, z in M.
Let (M.p) be a metric space. A mapping T from M into M is called a contraction if
for some constant a with 0 sa<1, and all a and y in M
I. Prove that the real line R becomes a metric space if we define a distance ρ(x, y) between
real numbers a and y by
Here, your objective is to show that p(z. g) satisfies the three conditions (i), (), and
(ii) above. You may assume without proof the "triangle" inequality for the absolute
value function
for all a, b in R
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metric space is a set M together with a distance function ρ(x,y) that represents the distance" between elements a and y of the set M. The distance function must satisfy ) plz,y)20 and plr,y) 0 if and only if y: (ii) p(x,y) o(y,z); (iii) p(z, y) a(x, z) +a(z, y) for all x, y, z in M. Let (M.p) be a metric space. A mapping T from M into M is called a contraction if for some constant a with 0 sa<1, and all a and y in M I. Prove that the real line R becomes a metric space if we define a distance ρ(x, y) between real numbers a and y by Here, your objective is to show that p(z. g) satisfies the three conditions (i), (), and (ii) above. You may assume without proof the "triangle" inequality for the absolute value function for all a, b in R

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Expert Answer

Step 1

Proof:

The metric defined on R is d (x, y) = |x – y|.

In order to prove the real line is a metric space with given metric, check the three conditions as follows.

To check for the non-negativity of the metric.

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Step 2

Check the symmetry property.

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Step 3

Check the triangle inequ...

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