???:Mathematical induction
???:Fundamental theorem of arithmetic (existence)
???:Irrationality of the square root of 2
There does not exist a rational number such that its square equals . This means that the real number is irrational.
???:Theorem of Euclid (prime numbers)
There exist infinitely many prime numbers.
???:Binomial theorem
Let be elements of a
field
and let denote a natural number. Then
-
holds.
???:Algebraic structure of the complex numbers
???:Euclidean division (polynomial ring)
???:Linear factor and zero of a polynomial
???:Number of zeroes of a polynomial
???:Fundamental theorem of algebra
Every nonconstant
polynomial
over the
complex numbers has a
zero.
???:Interpolation theorem for polynomials
Let be a
field,
and let different elements
and elements
are given. Then there exist a unique
polynomial
of degree , such that
holds for all .
???:Convergent sequence is ...
???:Rules for convergent sequences
Let
and
be
convergent sequences. Then the following statements hold.
- The sequence is convergent, and
-
holds.
- The sequence is convergent, and
-
holds.
- For
,
we have
-
- Suppose that
and
for all
.
Then is also convergent, and
-
holds.
- Suppose that
and that
for all
.
Then is also convergent, and
-
holds.
???:Real subset bounded from above
Every nonempty subset of the real numbers, which is
bounded from above, has a
supremum
in .
???:Cauchy criterion for series
Let
-
be a
series
of
real numbers. Then the series is
convergent
if and only if the following Cauchy-criterion holds: For every
there exists some such that for all
-
the estimate
-
holds.
???:Behavior of series members in case of convergence
Let
-
denote a
convergent
series
of
real numbers. Then
-
???:Leibniz criterion for alternating series
Let be an decreasing
null sequence
of nonnegative
real numbers. Then the
series
converges.
???:Absolute convergence and convergence
???:Direct comparison test