2. Cauchy criterion
2.1. Euclidean spaces (a brief review). Recall that a Euclidean space RN
is a linear space over real numbers whose elements are called vectors and they are ordered N−tuples of real numbers, x = (x1, x2, ..., xN), which
are called components. Two vectors are equal if they have identical components. The zero vector 0 has zero components. Two operations are defined for vectors, addition and multiplication by a scalar (real number), by the following rules:
x+ y = (x1+ y1, x2+ y2, ..., xN + yN)
αx = (αx1, αx2, ..., αxN) , α ∈ R
In addition, the space is equipped with a dot (scalar) product: (x, y) = x1y1+ x2y2+ · · · + xNyN
which is linear
(αx + βy, z) = α(x, z) + β(y, z) The norm or length of a vector is defined by
|x| = p(x, x) ≥ 0 The norm is equal to zero if and only if x = 0.
The scalar product satisfies the Cauchy-Schwartz inequality |(x, y)| ≤ |x||y|
from which the triangle inequality follows: |x| − |y| ≤ |x− y| ≤ |x| + |y|
The reader is advised to prove the above inequalities. A possible way to do so is to consider a real-valued quadratic function
f(t) = |x − ty|2
= |x|2
− 2t(x, y) + t2
|y|2
≥ 0
for any real t and any vectors x and y. In particular, f(t0) ≥ 0 where
t0 is the point at which f attains its absolute minimum. The
Cauchy-Schwartz inequality follows from the latter. The triangle inequality can be deduced from |x − y|2
≥ 0 and the Cauchy-Schwartz inequality. The norm defines a distance in a Euclidean space
|x − y| =p(x1− y1)2+ (x2 − y2)2+ · · · + (xN − yN)2
The distance function satisfies the distance axioms:
|x − y| ≥ 0 and |x − y| = 0 ⇔ x= y |x − y| = |y − x|
that is, it is non-negative, vanishes if and only if the elements are identical, is symmetric, and satisfies the triangle inequality.
In particular, a complex plane can always be viewed as R2
(every complex number z is a vector with components Re z and Im z). One writes C ∼ R2
.
2.2. Sequences in a Euclidean space. A sequence {xn}∞1 ⊂ RN is said
to converge to a vector x ∈ RN if
lim
n→∞|xn− x| = 0 .
It follows from the inequality |xjn− xj| ≤ |xn− x| ≤
N
X
i=1
|xin− xi| , ∀j = 1, 2, ..., N
where j and i enumerate components of vectors xn and x, that the
sequence {xn} converges if and only if the real sequences {xjn}, j =
1, 2, ..., N, of components converge: lim
n→∞|xn− x| = 0 ⇔ n→∞lim |xjn− xj| = 0 , j = 1, 2, ..., N
2.3. Bolzano’s theorem in a Euclidean space. Bolzano’s theorem also holds for sequences in Euclidean spaces. Naturally, sequences of the components of a bounded sequence in a Euclidean space are bounded:
|xn| ≤ M ⇒ |xjn| ≤ M , j = 1, 2, ..., N
Therefore, every bounded sequence in a Euclidean space has a limit point. The components of a limit point are limit points of bounded sequences of components of the vector sequence. If a sequence has just one limit point (a unique limit point), then it converges to that point. In particular, Bolzano’s theorem holds for complex sequences.
2.4. Cauchy sequences. To verify whether a sequence converges or di-verges, one has to come up with a number L to which terms of the sequence can get arbitrary close. To show that L is the limit, one has to the definition of the limit to demonstrate that terms of the sequence stay arbitrary close to L. For example, if zn= 1/n, then it is
straight-forward to find a good candidate for the limit: L = 0. Then it is not difficult to see that terms zn stay arbitrary close to 0 because
|zn| < ε , ∀n > N
where an integer N > 1/ε. What if such an “educated” guess for a possible value of the limit is hard to make, e.g., if terms of the sequence
zn are not known as an explicit function function of the index n. How
convergence or divergence of the sequence in question can be studied? Of course, there are some sufficient criteria for convergence, e.g., for monotonic bounded sequence, but they work only for special sequences. The answer is provided by the Cauchy criterion for convergence.
A sequence {zn} is called a Cauchy sequence if the distance |zk− zn|
can be made arbitrary small for all sufficiently large n and k. In other words, for any positive number ε > 0 and all positive integers p, one can find a positive integer m such that
∀ε > 0 ∃m : |zn+p− zn| < ε , ∀n ≥ m , ∀p > 0
The latter implies that lim
n→∞|zn+p− zn| = 0 , ∀p = 1, 2, ...
Clearly, every convergent sequence is a Cauchy sequence. If L is the limit then by the triangle inequality
|zn+p− zn| = |zn+p− L + L − zn| ≤ |zn+p− L| + |zn− L|
so that the right side converges to zero because zn→ L as n → ∞. It
turns out that the converse is also true, which is known as the Cauchy criterion for convergence:
A sequence of complex numbers converges if and only if it is a Cauchy sequence:
∃L ∈ C : lim
n→∞zn= L ⇔ {zn} is a Cauchy sequence
A proof of the converse (sufficiency) is based on the Bolzano theorem and contains three essential steps:
• Every Cauchy sequence is a bounded sequence;
• Every Cauchy sequence has a limit point (by Bolzano’s theo-rem);
• Every Cauchy sequence has a unique limit point
Let {zn} be a Cauchy sequence. The first assertion follows from that
there exists a positive integer m such that |zm+p− zm| < 1 , ∀p > 0
(put ε = 1 in the definition). For any two complex numbers a and b |a| − |b| ≤ |a − b| ≤ |a| + |b|
The reader is advised to prove this inequality (the right side is just the triangle inequality). Therefore
|zm+p| < |zm| + 1 , ∀p > 0
and, hence, a Cauchy sequence is bounded:
|zn| ≤ M , M = 1 + max{|z1|, |z2|, ..., |zm|}
By Bolzano’s theorem it has a limit point z. Suppose z and z0
are two distinct limit points of a Cauchy sequence. Then for any integers n, p, and q, the following inequality holds:
|z − z0
| = |z − zn+p+ zn+p− zn+ zn− zn+q+ zn+q− z0|
≤ |z − zn+p| + |zn+p− zn| + |zn+q− zn| + |z0− zn+q|
The second and third terms can be made arbitrary small for any p and q by choosing n large enough because the sequence is a Cauchy sequence. Having set n, one can always find p to make the fist term arbitrary small because z is a limit point, and, for the same reason, the last term can be made arbitrary small by choosing q. So, by adjusting n, p, and q, the right side of the inequality can be made smaller than any preassigned positive number, in particular, smaller than |z − z0
|, which is impossible unless z = z0
.
Now let us repeat the above in mathematical terms, fix ε > 0. Then there exists n such that |zn+p− zn| < ε and |zn+q− zn| < ε for
any p and q. Since z is a limit point, any neighborhood of z contains infinitely many terms of the sequence and, hence, there exists p such that |z − zn+p| < ε. Similarly, |z0 − zn+q| < ε for some q. Therefore
|z − z0
| < 4ε which is impossible because ε > 0 can be arbitrary small unless z = z0
.
2.5. Cauchy criterion in a Euclidean space. The concept of a Cauchy sequence is naturally extended to any Euclidean space. A sequence {xn} is called a Cauchy sequence if for any positive number ε > 0 and
all positive integers p, one can find a positive integer m such that ∀ε > 0 ∃m : |xn+p− xn| < ε , ∀n ≥ m , ∀p > 0
The Cauchy criterion holds in all Euclidean spaces
Theorem 2.1. A sequence in a Euclidean space converges if and and only if it is a Cauchy sequence.
A proof follows the same steps as in the case of real or complex se-quences. The reader is advised to repeat them for a general Euclidean
space. Note that the Cauchy criterion asserts that every Cauchy se-quence has a unique limit point in a Euclidean space. This means that every Euclidean space is complete. In fact, not every linear space equipped with a distance function is complete. Examples of such spaces will be studied later.
2.6. Exercises.
1. If (x, y) = x1y1 + x2y2 + · · · + xNyN is the dot (scalar) product
in RN and |x| = (x, x)1/2, prove the Cauchy-Schwartz inequality:
|(x, y)| ≤ |x||y|
2. Use the Cauchy-Schwartz inequality to prove the triangle inequality:
|x| − |y| ≤ |x− y| ≤ |x| + |y|