§l. Convergent sequences and series
59
3 - Local or asymptotic properties
As the reader will observe, one constantly has to examine functions of a single
variable - it may vary on an interval of JR, or only on N, or on some subset
of JR or of C, just to restrict ourselves to functions of a single real or complex
variable - when the variable is either "very close" to a fixed value a where
the function is or is not defined, or is "very large" i.e. "very close to infinity" .
One might like to know for example that x 2 is equal to 1 to within 0.00001
provided that x is "sufficiently close" to 1, that (2x + l)/(x - 1) is equal to
2 to within 0.001 provided that x is "sufficiently large" or that 1/n 3 is less
than 10- 100 when the positive integer n is "sufficiently large"; one also has
to know that, for x close to 0, x 3 is "negligible" in comparison to x, that for
n very large lO100 n 2 + lO100000n is "of the same order of magnitude" as n 2 ,
etc. It is easy to give a perfectly clear meaning to these a priori rather vague
expressions.
It is best to consider generally an assertion P(x) in which there appears
a letter x (or y, or n, or p, or some other) supposed to represent a number
varying in a given set E of real or complex numbers; for example, the relations
Ix 2 - 11 < 0.00001 where x E E = JR,
1(2x + l)/(x - 1) - 21::; 0.001 where x E E = JR - {I},
1/n 3 < 1/10 100 where nEE = N - {O},
n 2 < 10 100 n 2 + 10100000n ::; (10100 + 1) n 2 where nEE = z.
In the case of JR, we shall say that P(x) is true for all sufficiently large
positive x E E (or, for short, true for x large) if there exists a number M
such that, for x E E, the relation x> M implies P(x). There is an analogous
definition for x sufficiently large and negative. If one does not specify "positive" or "negative", then we mean that P(x) is true for Ixl > M. In the case
of C. where these inequalities have no meaning, one says that P( x) is true
for large x if there exists a number M > 0 such that Ixl > M implies P(x):
in other words if P(x) is true outside a sufficiently large disc.
Given a number a E JR or C, we say similarly that P(x) is true for all
x E E sufficiently close to a or, more briefly, that P(x) is true on a neighbourhood of a, if there exists a number r > 0 such that
(3.1)
{[d(a,x) < r] & (x E E)} ~ P(x);
in plain language: P(x) is true for all x E E such that Ix - al < r. Another
formulation: we call 10 an open ball with centre a any set B(a, r) defined by
10 The use of the word "ball" rather than the word "interval" (in the case of JR)
or of the word "disc" or "circle" (in the case of IC) is justified by the case of
functions of several variables, i.e. defined on a subset of a Cartesian space JRP.
59
3 - Local or asymptotic properties
As the reader will observe, one constantly has to examine functions of a single
variable - it may vary on an interval of JR, or only on N, or on some subset
of JR or of C, just to restrict ourselves to functions of a single real or complex
variable - when the variable is either "very close" to a fixed value a where
the function is or is not defined, or is "very large" i.e. "very close to infinity" .
One might like to know for example that x 2 is equal to 1 to within 0.00001
provided that x is "sufficiently close" to 1, that (2x + l)/(x - 1) is equal to
2 to within 0.001 provided that x is "sufficiently large" or that 1/n 3 is less
than 10- 100 when the positive integer n is "sufficiently large"; one also has
to know that, for x close to 0, x 3 is "negligible" in comparison to x, that for
n very large lO100 n 2 + lO100000n is "of the same order of magnitude" as n 2 ,
etc. It is easy to give a perfectly clear meaning to these a priori rather vague
expressions.
It is best to consider generally an assertion P(x) in which there appears
a letter x (or y, or n, or p, or some other) supposed to represent a number
varying in a given set E of real or complex numbers; for example, the relations
Ix 2 - 11 < 0.00001 where x E E = JR,
1(2x + l)/(x - 1) - 21::; 0.001 where x E E = JR - {I},
1/n 3 < 1/10 100 where nEE = N - {O},
n 2 < 10 100 n 2 + 10100000n ::; (10100 + 1) n 2 where nEE = z.
In the case of JR, we shall say that P(x) is true for all sufficiently large
positive x E E (or, for short, true for x large) if there exists a number M
such that, for x E E, the relation x> M implies P(x). There is an analogous
definition for x sufficiently large and negative. If one does not specify "positive" or "negative", then we mean that P(x) is true for Ixl > M. In the case
of C. where these inequalities have no meaning, one says that P( x) is true
for large x if there exists a number M > 0 such that Ixl > M implies P(x):
in other words if P(x) is true outside a sufficiently large disc.
Given a number a E JR or C, we say similarly that P(x) is true for all
x E E sufficiently close to a or, more briefly, that P(x) is true on a neighbourhood of a, if there exists a number r > 0 such that
(3.1)
{[d(a,x) < r] & (x E E)} ~ P(x);
in plain language: P(x) is true for all x E E such that Ix - al < r. Another
formulation: we call 10 an open ball with centre a any set B(a, r) defined by
10 The use of the word "ball" rather than the word "interval" (in the case of JR)
or of the word "disc" or "circle" (in the case of IC) is justified by the case of
functions of several variables, i.e. defined on a subset of a Cartesian space JRP.