APPENDIX B

A Review of Some Mathematical Concepts

## B.1 Limits of Real Number Sequences

Limit and Limit Points

A sequence of real numbers *x*_{n}, *n* ≥ 1, is said to converge to the *limit x* ε R if for every ε > 0, there exists an *n*_{ε} such that for all *n* > *n*_{ε}, |*x*_{n} − x| < ε i.e., no matter how small an ε > 0 we take, there is a point in the sequence (denoted by *n*_{e}) 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 *x*_{k}, *k* ≥ 1, viewed as a set, is bounded above, and is such that *x*_{k} ≤ *x*_{k}+_{1} (*x*_{k} is a nondecreasing sequence), then lim_{n→∞} *x*_{n} exists and ...