A Review of Some Mathematical Concepts
Limit and Limit Points
A sequence of real numbers xn, n ≥ 1, is said to converge to the limit x ε R if for every ε > 0, there exists an nε such that for all n > nε, |xn − x| < ε i.e., no matter how small an ε > 0 we take, there is a point in the sequence (denoted by ne) such that all elements of the sequence after this point are within ε of x (the proposed limit). A limit, if it exists, is clearly unique. This is written as
If xk, k ≥ 1, viewed as a set, is bounded above, and is such that xk ≤ xk+1 (xk is a nondecreasing sequence), then limn→∞ xn exists and ...