# Real analysis

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#1
What will be the supremum and infimum of the set { m/(|m|+n) : n- natural numbers, m- integers}?
0
2 years ago
#2
(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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2 years ago
#3
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.
0
2 years ago
#4
(Original post by Infinity ∞)
What will be the supremum and infimum of the set { m/(|m|+n) : n- natural numbers, m- integers}?
hint:

Try vary m and n looking at extremal cases. You should be able to check your answer over at Wolframalpha.com/
1
2 years ago
#5
It helps to notice that .

Also, try and and see what happens for large .
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