5) Show that the following functions define a metric on R.
(a) d(x, y) = |x − y|
(a) d(x, y) = |x − y|
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Since |x − y| = |y − x|, (M2) is satisfied. Note that
|x − y| = 0 ⇔ x − y = 0 ⇔ x = y,
(M1) is satisfied. To show (M3), we will make use of the inequality |a + b| ≤ |a| + |b| for any
a, b ∈ R. Indeed, we have
d(x, y) = |x − y| = |(x − z) + (z − y)| ≤ |x − z| + |z − y| = d(x, z) + d(z, y)
for any x, y, z ∈ R.
|x − y| = 0 ⇔ x − y = 0 ⇔ x = y,
(M1) is satisfied. To show (M3), we will make use of the inequality |a + b| ≤ |a| + |b| for any
a, b ∈ R. Indeed, we have
d(x, y) = |x − y| = |(x − z) + (z − y)| ≤ |x − z| + |z − y| = d(x, z) + d(z, y)
for any x, y, z ∈ R.