§ 1. Convergent sequences and series
95
too much for the capacities of the time; a legend, maybe apocryphal, but
significant, tells that, when an editor came to ask him for a paper, he just
handed him the topmost of the pile of his latest productions. With such habits
and inspired proofs, though a little or very false as we shall see on various
occasions, he would have had a lot of trouble nowadays. Of course, in our
time, he would have been educated by mathematicians more "serious" than
Johann Bernoulli and would have conformed to the rules of the corporation
as, in his time, he conformed in his private life, to those he had absorbed in
the puritan society of Bale. Happily there were, mainly in France, people to
advance other things than mathematics, hydraulics and artillery.
8 - Algebraic operations on limits
It is indispensable to know what happens when one performs simple algebraic
operations on sequences which tend to limits. The theory rests on a very few
results.
Theorem 1. Let (Un) and (vn ) be two convergent sequences, with limits u
and v. Then the sequences (un + vn ) and (unvn ) converge to u + v and uv.
If v =I- 0, then Vn =I- 0 for large n, and the sequence (un/vn ), defined for large
n, converges to u/v.
In other words,
lim(un + vn )
lim(unvn )
lim(un/vn )
== lim Un + lim Vn ,
(lim un).(lim vn),
(lim un)/(lim vn ) if lim Vn =I- o.
One can add the sums of series too - consider the partial sums:
Let us move on to the proof of Theorem 1.
Case of a sum. We need only write that
for n sufficiently large, each of two last differences is < r /2; the left hand side
is thus < r, qed.
Case of a product. This relies on the following lemma (the continuity of
the map (x, y) 1-+ xy of ([:2 into C):
Lemma. Let u and v be complex numbers. For every r > 0 there exists a
number r' > 0 such that the relations
(8.1)
lu' - ul < r' & lv' - vi < r' imply lu'v' - uvl < r.
95
too much for the capacities of the time; a legend, maybe apocryphal, but
significant, tells that, when an editor came to ask him for a paper, he just
handed him the topmost of the pile of his latest productions. With such habits
and inspired proofs, though a little or very false as we shall see on various
occasions, he would have had a lot of trouble nowadays. Of course, in our
time, he would have been educated by mathematicians more "serious" than
Johann Bernoulli and would have conformed to the rules of the corporation
as, in his time, he conformed in his private life, to those he had absorbed in
the puritan society of Bale. Happily there were, mainly in France, people to
advance other things than mathematics, hydraulics and artillery.
8 - Algebraic operations on limits
It is indispensable to know what happens when one performs simple algebraic
operations on sequences which tend to limits. The theory rests on a very few
results.
Theorem 1. Let (Un) and (vn ) be two convergent sequences, with limits u
and v. Then the sequences (un + vn ) and (unvn ) converge to u + v and uv.
If v =I- 0, then Vn =I- 0 for large n, and the sequence (un/vn ), defined for large
n, converges to u/v.
In other words,
lim(un + vn )
lim(unvn )
lim(un/vn )
== lim Un + lim Vn ,
(lim un).(lim vn),
(lim un)/(lim vn ) if lim Vn =I- o.
One can add the sums of series too - consider the partial sums:
Let us move on to the proof of Theorem 1.
Case of a sum. We need only write that
for n sufficiently large, each of two last differences is < r /2; the left hand side
is thus < r, qed.
Case of a product. This relies on the following lemma (the continuity of
the map (x, y) 1-+ xy of ([:2 into C):
Lemma. Let u and v be complex numbers. For every r > 0 there exists a
number r' > 0 such that the relations
(8.1)
lu' - ul < r' & lv' - vi < r' imply lu'v' - uvl < r.