Let A be a sSubset of a metric space A. Given r€A, define the distar from x to A to be dist(z, A) = inf{d(z, y) : y € A}.

Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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2.4.7. Let A be a subset of a metric space X. Given r E X, define the distance
from r to A to be
dist(r, A) = inf{d(r, y) : y E A}.
2.4 Closed Sets
61
Prove the following statements.
(a) If A is closed, then r E A if and only if dist(r, A) = 0.
(b) dist(r, A) < d(r, y) + dist(y, A) for all r, y E X.
(c) dist(r, A) – dist(y, A)| < d(x, y) for all r, y E X.
Additionally, show by example that it is possible to have dist(r, A) = 0 even
when z ¢ A.
II
Transcribed Image Text:2.4.7. Let A be a subset of a metric space X. Given r E X, define the distance from r to A to be dist(r, A) = inf{d(r, y) : y E A}. 2.4 Closed Sets 61 Prove the following statements. (a) If A is closed, then r E A if and only if dist(r, A) = 0. (b) dist(r, A) < d(r, y) + dist(y, A) for all r, y E X. (c) dist(r, A) – dist(y, A)| < d(x, y) for all r, y E X. Additionally, show by example that it is possible to have dist(r, A) = 0 even when z ¢ A. II
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