Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Important theorems
Suppose that for every natural number a statement is given. Suppose further that the following conditions are fulfilled.
- is true.
- For all we have: if holds, then also holds.
Every natural number , , has a factorization into prime numbers. That means there exists a representation
with prime numbers .
There does not exist a rational number such that its square equals . This means that the real number is irrational.
There exist infinitely many prime numbers.
Let be a field and let be the polynomial ring over . Let be polynomials with . Then there exist unique polynomials such that
Let be a field and let be the polynomial ring over . Let be a polynomial and . Then is a zero of if and only if is a multiple of the linear polynomial .
Let be a field and let be the polynomial ring over . Let be a polynomial () of degree . Then has at most zeroes.
Every nonconstant polynomial over the complex numbers has a zero.
Let be a field, and let different elements , and elements be given. Then there exists a unique polynomial of degree , such that holds for all .
Let and denote real sequences. Suppose that
and that and converge to the same limit . Then also converges to .
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.
Every nonempty subset of the real numbers, which is bounded from above, has a supremum in .
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.
Let be an decreasing null sequence of nonnegative real numbers. Then the series converges.
Let be a convergent series of real numbers and a sequence of real numbers fulfilling for all . Then the series
For all real numbers with , the geometric series converges absolutely, and the sum equals
Let
be a series of real numbers. Suppose there exists a real number with , and a with
for all (in particular for ). Then the series converges absolutely.