TITLE: What is the difference between a supremum and maximum; and also between the infimum and minimum?
QUESTION [18 upvotes]: What is the difference between a supremum and maximum; and similarly the infimum and minimum? Also, how does one tell if they exist?
Here is an example:

$$x_n = \frac{n}{2n-1}$$
Determine whether the maximum, the minimum, the supremum, and the
  infimum of the sequence $x_n$ n=1 to n=+∞

My understanding: 

The limit is 1/2. 
$x_n$ is decreasing. 
The supremum exists as n goes to +∞. 
The infimum does not exist as limited by n=1, minimum = 1/2. 
The maximum and supremum exists, both equal to 1.

REPLY [2 votes]: Apparently, you know how to obtain the limits (here: upper bound 1, lower bound 1/2). That is any case is then the supremum (or infimum at the lower end) - or the set is unlimited, e.g. the naturals on top, the reals both ways: Nothing exists then.
Two cases:
(1) The limit is IN the set, e.g. actually attained by the sequence.
Then it is maximum AND supremum (or minimum AND infimum), e.g. 1.
(2) The limit is OUTSIDE the set, not attained by the sequence (though only barely missed). That is your case at the lower end, 1/2.
Then it is only supremum (or infimum) AND there is no maximum (minimum).
You might have some doubts coming in via the index n (your 3rd suggestion $n\to\infty$ and supremum)?
$n\to\infty$  has nothing to do with supremum ("large $n$"?). Only the $x_n$ themselves do.
Take-home bits:

No bound - no ..mum, nothing.
Minimum = Infimum OR no Minimum. (ditto max=sup or no max). In particular, these can not differ!