58
II - Convergence: Discrete variables
To establish (II!'), one puts e = (a + b)/2, r = (b - a)/2, and any u E Q
satisfying Ie - ul < r will do.
It goes without saying that there always exist not just one, but infinitely
many, rational numbers between a and b: choose an x E Q between a and b,
then an x' E Q between a and x, then an x" E Q between a and x', etc.
In the sequel we shall constantly need to speak of intervals in the set lR
of real numbers. Given two numbers a and b, one denotes by
[a,b]
the interval
a S x S b,
[a,b[
the interval
a S x < b,
]a,b]
the interval
a < x S b,
]a,b[
the interval
a < x < b.
These intervals (which may be empty if a > b) differ one from the other
only in so far as they do, or do not, contain their endpoints. We shall also
sometimes employ a notation such as la, b) to denote an interval "open on
the left" but possibly open or closed on the right, optionally. For every real
number a, one also denotes by
[ a,+oo[
the interval
a S x,
] a,+oo[
the interval
a< x,
]- 00, a]
the interval
x S a,
]- oo,a[
the interval
x < a.
Finally, one sometimes denotes lR by the analogous notation] - 00, +00[.
The intervals of the form [a, b], [a, +00[, ] - 00, a] are called closed; those
of the form la, b[, with a and b possibly infinite, are called open, the whole
intervallR =] - 00, +oo[ being simultaneously open and closed. Finally, the
intervals of the form [a, b] with a and b finite are called compact. Later we
shall define much more general open, closed and compact sets.
As to the exact meaning of the symbols +00 and -00, used above or
elsewhere, one must be clear 9 that (my italics)
(1) 00 by itself means nothing, although phrases containing it sometimes
mean something,
(2) that in every case in which a phrase containing the symbol 00 means
something it will do so simply because we have previously attached a
meaning to it by means of a special definition.
9 Here I quote G. H. Hardy, Pure Mathematics (Cambridge University Press, 1908,
Tenth ed., 1963, p. 117); it would be difficult to put it better.
II - Convergence: Discrete variables
To establish (II!'), one puts e = (a + b)/2, r = (b - a)/2, and any u E Q
satisfying Ie - ul < r will do.
It goes without saying that there always exist not just one, but infinitely
many, rational numbers between a and b: choose an x E Q between a and b,
then an x' E Q between a and x, then an x" E Q between a and x', etc.
In the sequel we shall constantly need to speak of intervals in the set lR
of real numbers. Given two numbers a and b, one denotes by
[a,b]
the interval
a S x S b,
[a,b[
the interval
a S x < b,
]a,b]
the interval
a < x S b,
]a,b[
the interval
a < x < b.
These intervals (which may be empty if a > b) differ one from the other
only in so far as they do, or do not, contain their endpoints. We shall also
sometimes employ a notation such as la, b) to denote an interval "open on
the left" but possibly open or closed on the right, optionally. For every real
number a, one also denotes by
[ a,+oo[
the interval
a S x,
] a,+oo[
the interval
a< x,
]- 00, a]
the interval
x S a,
]- oo,a[
the interval
x < a.
Finally, one sometimes denotes lR by the analogous notation] - 00, +00[.
The intervals of the form [a, b], [a, +00[, ] - 00, a] are called closed; those
of the form la, b[, with a and b possibly infinite, are called open, the whole
intervallR =] - 00, +oo[ being simultaneously open and closed. Finally, the
intervals of the form [a, b] with a and b finite are called compact. Later we
shall define much more general open, closed and compact sets.
As to the exact meaning of the symbols +00 and -00, used above or
elsewhere, one must be clear 9 that (my italics)
(1) 00 by itself means nothing, although phrases containing it sometimes
mean something,
(2) that in every case in which a phrase containing the symbol 00 means
something it will do so simply because we have previously attached a
meaning to it by means of a special definition.
9 Here I quote G. H. Hardy, Pure Mathematics (Cambridge University Press, 1908,
Tenth ed., 1963, p. 117); it would be difficult to put it better.