Unizor - Algebra - Limits - Introduction
Channel: Zor Shekhtman on UNIZOR Education
Date: 2012-09-10

Intuitively, a sequence limit is a value, to which elements of a sequence are getting closer and closer as their order number is increasing to infinity. Let's try to define it more rigorously.
L is a limit of a sequence  a(n) , if for any d there exists order number N such that
|L - a(n)| is smaller than d for any n greater than N.
A sequence that has a limit is called convergent, it converges to its limit.
Theorem 1
A convergent sequence is bounded
Theorem 2
Limit of a sequence, multiplied by a factor, is equal to a limit of an original sequence, multiplied by this factor.
Theorem 3
Limit of a sum of two sequences is equal to a sum of their limits.
Theorem 4
Limit of a product of two sequences is equal to a product of their limits.
Theorem 5
Limit of an inverse of a sequence is equal to an inverse of its limit (assuming none is equal to zero).