# Minimum-so-far

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Suppose  is an arithmetic function from the natural numbers to a subring of the ring of real numbers. Then, a natural number  is termed a minimum-so-far for  if  for all natural numbers .
• Strict minimum-so-far is a natural number  such that  for .
• Maximum-so-far is a natural number  such that  for all .
• Strict maximum-so-far is natural number  such that  for all .