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.
infinitely small vs arbitrarily small
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real-analysis
real-numbers
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2well, what does infinitely small mean? – 2017-01-22
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00? or some value that tends to 0? – 2017-01-22
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1What does "value that tends to 0" mean? Can you give an example of a value that tends to 0? – 2017-01-22
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0"Arbitrarily small" is not a condition on a number. It's an implicit quantifier on what you're about to say. – 2017-01-22
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3When 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. – 2017-01-22
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0Yes, 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. – 2017-01-22
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2The 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. – 2017-01-22