???: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
???: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
-
![{\displaystyle {}\lim _{n\rightarrow \infty }{\left(x_{n}+y_{n}\right)}={\left(\lim _{n\rightarrow \infty }x_{n}\right)}+{\left(\lim _{n\rightarrow \infty }y_{n}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccaea0883f278be4e8840523b29c361eeaeffbd0)
holds.
- The sequence
is convergent, and
-
![{\displaystyle {}\lim _{n\rightarrow \infty }{\left(x_{n}\cdot y_{n}\right)}={\left(\lim _{n\rightarrow \infty }x_{n}\right)}\cdot {\left(\lim _{n\rightarrow \infty }y_{n}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fef7fa4d521a34a759b61acd5c5d3e9ae86fa4dd)
holds.
- For
,
we have
-
![{\displaystyle {}\lim _{n\rightarrow \infty }cx_{n}=c{\left(\lim _{n\rightarrow \infty }x_{n}\right)}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c51f600686e16716d9a442c06d9a5aa6f05a107e)
- Suppose that
and
for all
.
Then
is also convergent, and
-
![{\displaystyle {}\lim _{n\rightarrow \infty }{\frac {1}{x_{n}}}={\frac {1}{x}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9638896a75aaf6719bde39319af296495efd137)
holds.
- Suppose that
and that
for all
.
Then
is also convergent, and
-
![{\displaystyle {}\lim _{n\rightarrow \infty }{\frac {y_{n}}{x_{n}}}={\frac {\lim _{n\rightarrow \infty }y_{n}}{x}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94a7abb2a5d29b25c5885ca57802fbe26fd6bc9d)
holds.
???:Real subset bounded from above
Every nonempty subset of the real numbers, which is
bounded from above, has a
supremum
in
.
???:Bounded increasing real sequence ...
???: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
-
![{\displaystyle {}n\geq m\geq n_{0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be4378450dfdb522ccfe0d1063d2a3ea9894fb8d)
the estimate
-
![{\displaystyle {}\vert {\sum _{k=m}^{n}a_{k}}\vert \leq \epsilon \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50d2a70423d3b15009bc7032c504d61a12845733)
holds.
???:Behavior of series members in case of convergence
Let
-
denote a
convergent
series
of
real numbers. Then
-
![{\displaystyle {}\lim _{k\rightarrow \infty }a_{k}=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbdef0f3ee0837d98d8fffa66c0bd275be5c18fe)
???: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