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'''Supremum and infimum vs. maximum and minimum''' | '''Supremum and infimum vs. maximum and minimum''' | ||
| − | The concept of supremum, or least upper bound, is as follows: Let <math>S={a[n]}</math>, the sequence with terms <math>a[0],a[1],\cdots</math> over all the nonnegative integers. <math>S</math> has a supremum, called <math>\sup S</math> , if for every <math>n , a[n]\leq\sup S</math> (i.e. no a[n] exceeds <math>\sup S</math> ), and furthermore, <math>\sup S</math> is the least value with this property; that is, if <math>a[n]\leq b</math> 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. | + | The concept of supremum, or least upper bound, is as follows: Let <math>S={a[n]}</math>, the sequence with terms <math>a[0],a[1],\cdots</math> over all the nonnegative integers. <math>S</math> has a supremum, called <math>\sup S</math> , if for every <math>n , a[n]\leq\sup S</math> (i.e. no a[n] exceeds <math>\sup S</math> ), and furthermore, <math>\sup S</math> is the least value with this property; that is, if <math>a[n]\leq b</math> for all <math>n</math>, then <math>\sup S\leq b</math> for all such </math>b<math> . 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 <math>\inf S</math> , or greatest lower bound. |
Revision as of 11:17, 16 November 2010
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 </math>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 <math>\inf S $ , or greatest lower bound.
