98
II - Convergence: Discrete variables
§2. Absolutely convergent series
9 - Increasing sequences. Upper bound of a set of real numbers
For a start let us make some remarks on passing to the limit in inequalities,
essential for understanding axiom (IV) of nO 1.
First, it is obvious that if a sequence of numbers, positive for n large,
converges, then its limit is again positive; the reader may provide the c and
the N necessary for a textbook proof ... It follows from this that
(9.1)
a::; un::; b for all n large ===} a ::;limun ::; b,
as one sees on considering the sequences Un - a and b - Un. Similarly
(9.2)
Un ::; Vn for all n large ===} lim Un ::; lim Vn
since the terms of the sequence Vn - Un are positive for n large.
In other words, weak inequalities are preserved under passage to the limit.
Not so for strict inequalities: one has lin> 0 for all n, but lim lin = o.
Without explicit evidence to the contrary, passing to the limit transforms
strict inequalities into weak inequalities: the inequality lin> -2 is preserved
in the limit since there exists a number r > 0 such that lin> -2 + r for all
n, so that the limit is ~ -2 + r > -2.
These results, though trivial, bring us back to axiom (IV) for 1R mentioned
in nO 1 of this chapter. Consider an increasing sequence
(9.3)
of real numbers. For a sequence to converge, it is clearly necessary that,
increasing or not, there exists a positive number M such that Iunl ::; M for
all n, i.e. that the sequence should be bounded. In the case of interest this
means the existence of numbers M which majorise the sequence, i.e. satisfy
M ~ Un for all n, the weak inequality being essential in what follows. One
also says that M is a majorant of the given sequence and that the latter is
majorised by M, or majorised for short if one does not want to specify M
exactly, or also bounded above.
Suppose now that the sequence (3) converges to a limit u. By property
(1) above, the relation
(9.4)
Up 5 M (for all p) implies u 5 M.
Since on the other hand up ::; Un for all n ~ p, one sees similarly, on passing
to the limit over n, that also
(9.5)
Up 5 U for all p.
The relation (4) shows that every majorant is 2: u, and (5) that U is one of
these majorants; conclusion:
II - Convergence: Discrete variables
§2. Absolutely convergent series
9 - Increasing sequences. Upper bound of a set of real numbers
For a start let us make some remarks on passing to the limit in inequalities,
essential for understanding axiom (IV) of nO 1.
First, it is obvious that if a sequence of numbers, positive for n large,
converges, then its limit is again positive; the reader may provide the c and
the N necessary for a textbook proof ... It follows from this that
(9.1)
a::; un::; b for all n large ===} a ::;limun ::; b,
as one sees on considering the sequences Un - a and b - Un. Similarly
(9.2)
Un ::; Vn for all n large ===} lim Un ::; lim Vn
since the terms of the sequence Vn - Un are positive for n large.
In other words, weak inequalities are preserved under passage to the limit.
Not so for strict inequalities: one has lin> 0 for all n, but lim lin = o.
Without explicit evidence to the contrary, passing to the limit transforms
strict inequalities into weak inequalities: the inequality lin> -2 is preserved
in the limit since there exists a number r > 0 such that lin> -2 + r for all
n, so that the limit is ~ -2 + r > -2.
These results, though trivial, bring us back to axiom (IV) for 1R mentioned
in nO 1 of this chapter. Consider an increasing sequence
(9.3)
of real numbers. For a sequence to converge, it is clearly necessary that,
increasing or not, there exists a positive number M such that Iunl ::; M for
all n, i.e. that the sequence should be bounded. In the case of interest this
means the existence of numbers M which majorise the sequence, i.e. satisfy
M ~ Un for all n, the weak inequality being essential in what follows. One
also says that M is a majorant of the given sequence and that the latter is
majorised by M, or majorised for short if one does not want to specify M
exactly, or also bounded above.
Suppose now that the sequence (3) converges to a limit u. By property
(1) above, the relation
(9.4)
Up 5 M (for all p) implies u 5 M.
Since on the other hand up ::; Un for all n ~ p, one sees similarly, on passing
to the limit over n, that also
(9.5)
Up 5 U for all p.
The relation (4) shows that every majorant is 2: u, and (5) that U is one of
these majorants; conclusion: