§3. Bolzano-Weierstrass and Cauchy's criterion
231
of three dimensions one knows, for example, that if F is a plane or a line,
then the point of F at the minimum distance to x is precisely the orthogonal
projection of x onto F.
Cauchy's criterion extends naturally to functions of a "non-discrete" variable:
Theorem 13'. Let X be a subset ofC, let a be an adherent point of X and
f a scalar function defined on X. For f to tend to a limit as x E X tends
to a it is necessary and sufficient that for all r > 0 there exists an r' > 0
such that, for x', x" EX,
(10.2)
x', x" E B(a, r') ===} d[f(x'),J(x")] < r.
If 11. is the limit value of f at a then d[f(x), u] ::; r for d(a, x) < r'.
, There is an analogous result when, X c lR not being bounded above, one
lets x tend to +00: for all r > 0 there must exist an M such that
{(x' > M) & (x" > M)} ===} If(x') - f(x") I < r.
r The proof of Theorem 13' is practically identical to that of Cauchy's
criterion for sequences; on the other hand one can deduce the former from
,Theorem 13' by choosing X = Nand f(x) = Un for x = n. Condition (2)
then means that lup - uql < r for p and q sufficiently large. But it is simpler
to deduce Theorem 13' from Theorem 13. To do this one chooses a sequence
Of points Xn E X tending to a; with the notation of (2), the ball (open or
closed, it doesn't matter) B(a, r') with centre a and radius r' contains xp and
:l;q for p, q large, whence
tlO.3)
d[f(xp), f(x q)] < r;
io the points f(xn ) form a Cauchy sequence, so tend to a limit u. But for
x E B(a, r') and n large enough that Xn E B(a, r'), f(x) is equal to f(xn )
. to within r, so equal to u to within 2r, whence convergence. The inequality
d[f(x),u] ::; r for d(x,a) < r' follows from observing that d[f(x),J(y)] < r
for all y E B(a, r') and passing to the limit 19 when y tends to a.
Finally, there is a Cauchy criterion for uniform convergence. Here we use
the concept of the distance between two functions as defined in nO 7:
(lOA)
dx(f,g) = sup If(x) - g(x)l·
xEX
19 The remarks at the beginning of nO 9 of Chap. II on limits of inequalities clearly
apply also to functions more general than sequences.
231
of three dimensions one knows, for example, that if F is a plane or a line,
then the point of F at the minimum distance to x is precisely the orthogonal
projection of x onto F.
Cauchy's criterion extends naturally to functions of a "non-discrete" variable:
Theorem 13'. Let X be a subset ofC, let a be an adherent point of X and
f a scalar function defined on X. For f to tend to a limit as x E X tends
to a it is necessary and sufficient that for all r > 0 there exists an r' > 0
such that, for x', x" EX,
(10.2)
x', x" E B(a, r') ===} d[f(x'),J(x")] < r.
If 11. is the limit value of f at a then d[f(x), u] ::; r for d(a, x) < r'.
, There is an analogous result when, X c lR not being bounded above, one
lets x tend to +00: for all r > 0 there must exist an M such that
{(x' > M) & (x" > M)} ===} If(x') - f(x") I < r.
r The proof of Theorem 13' is practically identical to that of Cauchy's
criterion for sequences; on the other hand one can deduce the former from
,Theorem 13' by choosing X = Nand f(x) = Un for x = n. Condition (2)
then means that lup - uql < r for p and q sufficiently large. But it is simpler
to deduce Theorem 13' from Theorem 13. To do this one chooses a sequence
Of points Xn E X tending to a; with the notation of (2), the ball (open or
closed, it doesn't matter) B(a, r') with centre a and radius r' contains xp and
:l;q for p, q large, whence
tlO.3)
d[f(xp), f(x q)] < r;
io the points f(xn ) form a Cauchy sequence, so tend to a limit u. But for
x E B(a, r') and n large enough that Xn E B(a, r'), f(x) is equal to f(xn )
. to within r, so equal to u to within 2r, whence convergence. The inequality
d[f(x),u] ::; r for d(x,a) < r' follows from observing that d[f(x),J(y)] < r
for all y E B(a, r') and passing to the limit 19 when y tends to a.
Finally, there is a Cauchy criterion for uniform convergence. Here we use
the concept of the distance between two functions as defined in nO 7:
(lOA)
dx(f,g) = sup If(x) - g(x)l·
xEX
19 The remarks at the beginning of nO 9 of Chap. II on limits of inequalities clearly
apply also to functions more general than sequences.