What is the difference between minimum and infimum? What is the difference between minimum and infimum?
I have a great confusion about this.
 A: The minimum is attained, the infimum isn't necessarily. 
Example.
Let $f(x) = \frac{1}{x}$. Then $f$ has no minimum value on the interval $(0,\infty)$. The minimum is the smallest element in the set. That is
$$
\min\{f(x)\mid x\in (0,\infty)\}
$$
doesn't exist because there is not smallest number in the set.
Another example is the minimum of the set $S = (0,1) = \{x\mid 0<x<1\}$. Here again there isn't a smallest number
$$
\min\{x\mid 0<x<1\}
$$
doesn't exist.
The infimum of a set $S$ is defined as the greatest number that is less than or equal to all elements of S (from Wikipedia). The infimum is also sometimes called the greatest lower bound.
It is a fact that every non empty set (bounded below) of real numbers has an infimum. But, as we saw, not every real set has a minimum.
So in the example
$$
\inf\{f(x)\mid x\in (0,\infty)\} = 0.
$$
Note that the infimum and the minimum can be the same. Consider for example $S = \{1,2,3,\dots\}$. Then the infimum and minimum is both $1$.
Consider this other example. If $f$ is a continuous function on a closed interval $[a,b]$,  then it is a fact that $f$ attains a minimum over that interval. So here again
$$
\inf\{f(x)\mid x\in [a,b]\} = \min\{f(x)\mid x\in [a,b]\}.
$$
A: minimum is reached, infimum (may) not.
That is, the numbers of the form $1/n$ have an inf (that is, 0), while the natural numbers have a min (that is, 1).
