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II - Convergence: Discrete variables
The crucial point masked by this statement is the postulate that, among all
the majorants of the given sequence, there exists a number smaller than all
the others.
There is a similar statement for decreasing sequences: such a sequence
converges if and only if it is minorised, i.e. if there exist numbers m less than
all its terms. Its limit is then the largest of these minorants i.e. the greatest
lower bound of the set of its terms.
To clarify the reader's thoughts, it is indispensable to introduce, or to revise more systematically than we have done so far, some very easy definitions
in constant use.
One says that a set E c lR is bounded above , or majorised, if there exists
a number M such that x :s: M for all x E E; one then says that M majorises
E, or is a majorant of E, or that E is majorised by M. There are analogous
definitions for sets bounded below, or minorised, and numbers which minorise
the set, etc. Finally, one says that a set E c C is bounded if there exists a
number M ~ 0 such that Ixl :s: M for all x E E.
Let E c lR be a set bounded above and let M and M' be two majorants
of E. If M < M', the relation {x E E =* x :s: M} is clearly stronger than
the similar relation with M'. For example, it is not without interest to know
that, of the human species, everyone dies before attaining the age of 500
years, but to avoid surprises it is better to know that everyone dies before
250 years. This information again not being, it would seem, the best possible,
one might try to determine as small an age as possible before which everyone
dies. This would be the precise least upper bound of human life. Whence the
concept of the least upper bound of a set E c lR bounded above: it is a
number u = sup (E) satisfying the two following conditions:
(SUP 1)
(SUP 2)
x :s: u for all x E E, i.e. u majorises E;
u :s: M for any other majorant M of E.
In other words, sup(E) is the least majorant of E.
One could replace (SUP 2) by
(SUP 2') for all r > 0 there exists an x E E such that u - r < x.
If indeed u is the least possible majorant, then the number u - r does not
majorise E, whence the existence of x. If conversely (SUP 2') holds, then
every M majorising E majorises, for any r > 0, an x > u - r, so majorises
u - r for any r > 0, so majorises u (modified Archimedes' axiom), whence
(SUP 2).
(SUP 2') also implies that u is the limit of a sequence of elements of E:
choose Xn E E such that u - lin < x n ·
As we have seen above, the difficulty in proving that an increasing sequence tends to a limit disappears the moment the existence of the least
II - Convergence: Discrete variables
The crucial point masked by this statement is the postulate that, among all
the majorants of the given sequence, there exists a number smaller than all
the others.
There is a similar statement for decreasing sequences: such a sequence
converges if and only if it is minorised, i.e. if there exist numbers m less than
all its terms. Its limit is then the largest of these minorants i.e. the greatest
lower bound of the set of its terms.
To clarify the reader's thoughts, it is indispensable to introduce, or to revise more systematically than we have done so far, some very easy definitions
in constant use.
One says that a set E c lR is bounded above , or majorised, if there exists
a number M such that x :s: M for all x E E; one then says that M majorises
E, or is a majorant of E, or that E is majorised by M. There are analogous
definitions for sets bounded below, or minorised, and numbers which minorise
the set, etc. Finally, one says that a set E c C is bounded if there exists a
number M ~ 0 such that Ixl :s: M for all x E E.
Let E c lR be a set bounded above and let M and M' be two majorants
of E. If M < M', the relation {x E E =* x :s: M} is clearly stronger than
the similar relation with M'. For example, it is not without interest to know
that, of the human species, everyone dies before attaining the age of 500
years, but to avoid surprises it is better to know that everyone dies before
250 years. This information again not being, it would seem, the best possible,
one might try to determine as small an age as possible before which everyone
dies. This would be the precise least upper bound of human life. Whence the
concept of the least upper bound of a set E c lR bounded above: it is a
number u = sup (E) satisfying the two following conditions:
(SUP 1)
(SUP 2)
x :s: u for all x E E, i.e. u majorises E;
u :s: M for any other majorant M of E.
In other words, sup(E) is the least majorant of E.
One could replace (SUP 2) by
(SUP 2') for all r > 0 there exists an x E E such that u - r < x.
If indeed u is the least possible majorant, then the number u - r does not
majorise E, whence the existence of x. If conversely (SUP 2') holds, then
every M majorising E majorises, for any r > 0, an x > u - r, so majorises
u - r for any r > 0, so majorises u (modified Archimedes' axiom), whence
(SUP 2).
(SUP 2') also implies that u is the limit of a sequence of elements of E:
choose Xn E E such that u - lin < x n ·
As we have seen above, the difficulty in proving that an increasing sequence tends to a limit disappears the moment the existence of the least