Expert answer:real analysis (mathematics))))))
Answer & Explanation:real analysis _HW4.pdf
real_analysis__hw4.pdf
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MTH 430: Homework 4
1. Let (X, d) be a metric space. Prove that a subset S ⊆ X is bounded if and only if
the set {d(p, q) | p, q ∈ S} is bounded.
2. Let (X, d) be a metric space. Prove that (X, d1 ) is a metric space where d1 (x, y) =
[d(x, y)]2 .
3. Let d1 and d2 be two different metrics for the set X. Prove that d(x, y) = max(d1 (x, y), d2 (x, y))
is also a metric for X.
4. Let d(x, y) =
1
x
−
1
y
. Prove that d is a metric for the set X = (0, ∞).
5. Find the set B(10, .01) for the metric space in the previous problem.
6. Let d(P, Q) = max(|xp − xq |, |yp − yq |). Prove that d is a metric for the set R2 .
0
, 1/2 for the metric space in the previous problem.
7. Find the set B
0
1
…
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