Apples are on sale at the grocery store at a price of 8 apples for $4.00. Maggie buys 12 apples. If she is charged the sale pric
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By definition, if is the least upper bound of the set
, it means two thing:
In other words, the least upper bound of a set is greater than or equal to every single element of the set, but it is "close enough" to the elements of the set, because you guaranteed to find elements in the set between and
For example, pick . Obvisouly, the least upper bound is
. In fact, every number in
is smaller than 10, but as soon as you take away something from 10, say 0.01, you get 9.99, and there are elements in
greater than 9.99, say 9.9999.
So, the claim is basically proven by definition: if , let
. By definition, there exists
.