9780023606922, Chapter 2 1, Problem 63E

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Chapter 2.1, Problem 63E Step-by-step solution Step 1 1 of 4 Greatest lower bound of a non-empty set Now going to prove that "every non-empty set which is bounded below has a greatest lower bound", if "every non-empty set which is bounded above has a least upper bound". Assume X is a non-empty set of real numbers, a is an upper bound if for a is said to be the least upper bound of X, if d is also an upper bound of X so that, Step 2 of 4 Now, a is said to be a lower bound of X, if for all So, a is said to be the greatest lower bound of if b is also a lower bound of X such that Step 3 of 4 Now assume, Y be a non-empty set which is bounded above and also bounded below, that is, Consider the following graph for set d a f y b e In the above graph, a is the least upper bound because ab and y>c, where b and are lower bounds of Y. Then b will be the greatest lower bound if will be the greatest lower bound if c>b. Hence there exists a greatest lower bound. Therefore, if every non-empty set which is bounded above has a least upper bound, then every non-empty set which is bounded below has a greatest lower bound.