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I'd like some help with finding the supremum and infimum of $$\left\{ 2a,3(1-2a),5a:0<a<\frac{1}{2}\right\} $$
My guess is that the infimum is 0 and the supremum is 3.

Thanks.

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All 3 functions are continuous and defined on the closed interval [0,1/2] so all 3 have maxes on [0, 1/2] and must occur at a place where derivative is 0, derivative is undefined, or end point of interval. The derivatives are only constant so max and min of each function must occur at end points. So just plug in 0 and 1/2 to all 3 functions. The smallest of these must be your min (over all 3 functions) and the biggest your max (over all 3 functions). And, if you take away those 2 endpoints now, you won't get those values any more but you will get arbitrarily close. And, you don't get anything below 0 or anything above 3, so you are correct.

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Thanks for your explanation and confirming my answer. –  fav Apr 9 '11 at 21:09

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