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difference between MAXIMUM and SUPREMUM

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darock

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can anyone tell me the difference between MAXIMUM and SUPREMUM?

similarly minimum and infimum..
i tried online but couldnt find any
 

An example here.

Consider a set X={x:x<2}

there is no max{X}, but sup{X}=2
 

    darock

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thanks..nice ex.

Added after 2 minutes:

Actually, why do you say there is no maximum in the region (0,2)?

I understand 2-\epsilon is the max. I am wondering how to solve optimization problems when the solution lies in the boundary and when the boundary point is not included?
 

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