the field of complex numbers is algebraically closed

any integer > 1 can be written as a unique product of primes

for any prime p which does not divide the integer a, then the (p-1)st power of a leaves remainder 1 upon division by p

the order of any subgroup of a finite group G divides the order of G

a bounded sequence of real numbers has a convergent subsequence

If f is a real-valued continuous function on the interval [a, b], and u is a number between f(a) and f(b), then there is a c ∈ [a, b] such that f(c) = u

no consistent system of axioms whose theorems can be listed by an 'effective procedure' (essentially, a computer program) is capable of proving all facts about the natural numbers.

If A is any set, then there is no surjection of A onto the power set of A, the set of all subsets of A

a monotone sequence of real numbers is convergent if and only if it is bounded

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