1.1 Basic Mathematics

1.1.1 Mathematical notation

• ≈ : approximately equal

• ~ : CST ·

Supremum and infimum vs. maximum and minimum

The concept of supremum, or least upper bound, is as follows: Let S={a[n]} , the sequence with terms a[0],a[1],\cdots over all the nonnegative integers. S has a supremum, called \sup S , if for every n , a[n]\leq\sup S (i.e. no a[n] exceeds \sup S ), and furthermore, \sup S is the least value with this property; that is, if a[n]\leq b for all n, then \sup S\leq b for all such b . This is why the supremum is also called the least upper bound, for a bound is a number which a function, sequence, or set, never exceeds. Similarly, one can define the infimum \inf S , or greatest lower bound.