Infinity...
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What will be the supremum and infimum of the set { m/(|m|+n) : n- natural numbers, m- integers}?
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Prasiortle
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(Original post by Infinity ∞)
What will be the supremum and infimum of the set { m/(|m|+n) : n- natural numbers, m- integers}?
Divide into cases for when m is positive or negative, so that you can deal with the absolute value. I can't give you the full solution as the forum rules prohibit it, but hopefully this should get you started.
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square_peg
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Think about how n changes the expression m/(|m|+n). Try varying it to extremes.

It's really helpful in these problems when one of the numbers changes the expression monotonically. Is the function f(n) = m/(|m|+n) monotonic over the natural numbers? This helps us to find the extreme values. Notice though that this will be different depending on whether m is positive or negative.
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Ryanzmw
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(Original post by Infinity ∞)
What will be the supremum and infimum of the set { m/(|m|+n) : n- natural numbers, m- integers}?
hint: m > 0 :  \dfrac{m}{\vert{m}\vert + n} =  1- \dfrac{n}{ m+ n}
 m < 0: \dfrac{m}{\vert{m}\vert + n} = -1 - \dfrac{n}{m-n}

Try vary m and n looking at extremal cases. You should be able to check your answer over at Wolframalpha.com/
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ciberyad
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It helps to notice that \left|\dfrac m{|m|+n}\right|=\dfrac{|m|}{|m|+  n}\leq\dfrac{|m|}{|m|+1}<1.

Also, try m=n^2 and m=-n^2 and see what happens for large n.
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