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In a question I recently asked, Definitions of supremum it was pointed out that my usage of infinitly small was misguided, and the proper terminology was arbitrarily small. Could someone explain what is the difference between the two? Say in a context such as: take some $\epsilon$ infinitely small/arbitrary.

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rannoudanames
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    well, what does infinitely small mean? – operatorerror Jan 22 '17 at 20:40
  • 0? or some value that tends to 0? – rannoudanames Jan 22 '17 at 20:42
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    What does "value that tends to 0" mean? Can you give an example of a value that tends to 0? – Ittay Weiss Jan 22 '17 at 20:44
  • "Arbitrarily small" is not a condition on a number. It's an implicit quantifier on what you're about to say. – Qiaochu Yuan Jan 22 '17 at 20:44
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    When you declare $\epsilon$ to be infinitely small, you have assigned it a value already. When you say "let $\epsilon$ be arbitrarily small", it us up to the reader to choose a value for $\epsilon$. The latter definition is taken when you must prove something for any possible value. – David Jan 22 '17 at 20:45
  • Yes, the question from @Ittay Weiss makes me realise the inaccurate description of infinitely small. When arbitrarily small does not take some fixed value. I think i understand why we use the term arbitrary. – rannoudanames Jan 22 '17 at 20:49
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    The infinite is not a quantity in the reals, but when you says "$\epsilon$ is arbitrarily small" you are talking of quantities in the positive reals. – Masacroso Jan 22 '17 at 20:53

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