Increasing sequences have the pleasing property
that their convergence (or divergence) is easy to identify:
An increasing sequence converges if, and only
if, it is bounded above.
(See
BOUND
and
CONVERGENT SEQUENCE
.) This seems
clear geometrically. If on the real number line the
sequence of numbers {an} moves steadily to the right yet
does not penetrate a barrier at position B, then the
numbers must “pile up” at some limit value L≤B.
Conversely, if an increasing sequence converges to a
value L, then all the terms of the sequence lie to the left
of Lon the number line and hence are bounded above
by L. (These statements can be proved rigorously by
making use of the “completeness” of the real number
line via the notion of a D
EDEKIND CUT
.)
A sequence {an} is said to be decreasing if a1≥a2≥
a3≥…, that is, each term in the sequence is less than or
equal to the one that precedes it. It is called strictly
decreasing if a1> a2> a3>…. Similarly one can show
that a decreasing sequence converges if, and only if, it
is bounded below. A sequence that is either increasing
or decreasing is called monotonic.
A real-valued function fis said to be increasing
over an interval if, over that interval, greater input
values produce greater (or possibly equal) output val-
ues, that is, if aand bare two values in the interval
with a<b, then f(a) ≤f(b).If it is always the case
that f(a)<f(b),then the function is said to be strictly
increasing on the interval. The
MEAN
-
VALUE THEO
-
REM
shows that a differentiable function fis increas-
ing precisely on those intervals where the
DERIVATIVE
is nonnegative, f′(x) ≥0, and strictly increasing if
f′(x) > 0.
A real-valued function fis said to be decreasing
over an interval if greater input values produces smaller
(or possibly equal) output values of the function, that is,
if a< b, then f(a) ≥f(b).If it is always the case that f(a)
> f(b),then the function is said to be strictly decreasing
on the interval. The
MEAN
-
VALUE THEOREM
shows that
a differentiable function fis decreasing precisely on
those intervals for which f′(x) ≤0, and strictly decreas-
ing if f′(x) < 0.
A function that is either increasing or decreasing
on an interval is called monotonic on that interval.
See also
DIFFERENTIAL CALCULUS
;
GRAPH OF A
FUNCTION
;
MAXIMUM
/
MINIMUM
.
increment A small finite change in the value of a vari-
able or in the value of a function is called an increment.
If the variable is x, then an increment of xis usually
denoted ∆xor δx. If fis a function of x, then the corre-
sponding increment of f, denoted ∆f, equals ∆f= f(x+
∆x) – f(x).The
LIMIT
of the ratio ∆f/∆x, as the incre-
ment ∆xdecreases to zero, is called the
DERIVATIVE
of f.
A negative increment is sometimes called a decrement.
See also
DIFFERENTIAL
.
indefinite integral See
ANTIDIFFERENTIATION
;
INTE
-
GRAL CALCULUS
.
independent axiom An
AXIOM
of a mathematical
theory is said to be independent if it cannot be derived
from the remaining axioms of a theory as a logical con-
sequence. For instance, the existence of
NON
-
EUCLIDEAN GEOMETRIES
establishes that Euclid’s
famous
PARALLEL POSTULATE
is independent of his four
remaining postulates. In general, this is the approach
mathematicians take to show that an axiom Ain a
mathematical system is independent of the remaining
axioms: present an example of another consistent
mathematical theory proved to be free of contradic-
tions in which all axioms except Ahold and for which
Ais false. (Consequently, it cannot be the case that
these axioms do imply that Ais true, for then one has a
system in which both Aand not Ahold.)
See also E
UCLID
’
S POSTULATES
.
independent events Two experiments run in succes-
sion are deemed independent if the outcomes obtained
from one experiment do not affect the outcomes
obtained in the second. For example, the results from
casting a die do not influence the results obtained in
later tossing a coin. These two actions are independent.
In
PROBABILITY
theory we say two
EVENT
s Aand B
are independent if knowledge of Ahaving occurred has
no influence on the likelihood of Bnext occurring. For
example, in two rolls of die casting, an even number on
the first roll and casting a 6 on the second are indepen-
dent events—the chances of rolling a six are 1/6, no
matter the result of the first roll.
Two events that are not independent are called
dependent. For example, the event “Sally is wearing a
262 increment