Corollary 1.
If {a_{n}} and {b_{n}} are sequences with the
property that there exists some integer N such that a_{n} = b_{n} for all n > N then {a_{n}} is convergent if and only if {b_{n}} is convergent.
If the sequences are convergent then their limits are the same.
Theorem.
If there exists a function f which is defined for all real numbers x > 1 such that a_{n} = f(n) for all
positive integers n and such that the limit