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Let A be a set in a metric space and {U_i} be an open cover of A. A number r > 0 such that for all y in A, B(y,r) [itex]\subset[/itex] U_i for some i is calledaLebesgue number for the covering. The infimum of all Legesgue number is calledtheLebesgue number for the covering.

Am I wrong in thinking that if r is a Lebesgue number, then any other number lesser than r is also a Lebesgue number, so that if soon as a Lebesgue number exists, the Lebesgue number for the covering is 0?

After all, if for some r > 0, B(y,r) [itex]\subset[/itex] U_i for some i, then if r > a > 0, B(y,a) [itex]\subset[/itex] B(y,r) [itex]\subset[/itex] U_i, so that a is also a Lebesgue number.

:grumpy:

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# Homework Help: Analysis - Help me understand this dfn

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