Bounds If for all numbers x of a set there is a number M such that x s M, the set is bounded above and M is called an upper bound. Similarly if x 2 m, the set is bounded below and m is called a lower bound. If for all x we have m s xs M, the set is called bounded. If M is a number such that no member of the set is greater than M but there is at least one member which exceeds M - e for every e >0, then M is called the least upper bound (l.u.b.) of the set. Similarly, if no mem- ber of the set is smaller than m +e for every e >0, then m is called the greatest lower bound (g.l.b.) of the set.
Bounds If for all numbers x of a set there is a number M such that x s M, the set is bounded above and M is called an upper bound. Similarly if x 2 m, the set is bounded below and m is called a lower bound. If for all x we have m s xs M, the set is called bounded. If M is a number such that no member of the set is greater than M but there is at least one member which exceeds M - e for every e >0, then M is called the least upper bound (l.u.b.) of the set. Similarly, if no mem- ber of the set is smaller than m +e for every e >0, then m is called the greatest lower bound (g.l.b.) of the set.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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is the textbook missing a line?
should this statement be inserted? "if no member of the set is smaller than m, but there is at least one member that is smaller than....
why or why not?
Transcribed Image Text:Bounds
If for all numbers x of a set there is a number M such that x < M, the set is bounded above and M is called
an upper bound. Similarly if x > m, the set is bounded below and m is called a lower bound. If for all x we
have m < x < M, the set is called bounded.
If M is a number such that no member of the set is greater than M but there is at least one member which
exceeds M - e for every e>0, then Mis called the least upper bound (l.u.b.) of the set. Similarly, if no mem-
ber of the set is smaller than m +e for every e > 0, then m is called the greatest lower bound (g.l.b.) of the
set.
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