Definitions: max, min, sup and inf of a sequence I find it hard knowing the difference between the supremum of a series and and the infimum
So we have 
$\sup(a_n)$ and $\inf(a_n)$
what does it signify when we say a series has a maximum but not a minimum  or has both maximum and minimum etc?


*

*For example


$$a_n = 1 + 1/n$$
we know that the series is bounded by $1≤a_n≤2$, I know that it has a max which is 2 but
how come we say it doesn't have a minimum
 A: The set $[0, 1]$ contains all of the numbers between $0$ and $1$ including $0$ and $1$. Therefore, it has maximum element $1$ and minimum element $0$.
The set $[0, 1)$ contains all of the numbers between $0$ and $1$ including $0$ but not including $1$. Therefore, it has no maximum element, as if $m$ was the maximum element then $m<1$ and so $\frac{1+m}{2}\in[0, 1)$. However, $m<\frac{1+m}{2}<1$, which contradicts the maximality of $m$. But clearly this set has a "biggest element" (namely $1$), the twist is that this element isn't actually in the set. So we give this "maximum if it was in the set" element a name, "the supremum", and give it a formal definition (the least element greater than or equal to all elements of $[0, 1)$, or whichever set you are working with).
The concept of infimum is analogous, and this can all be applied to your specific example.
A: Minimum is infimum when infimum is an element of the set, here infimum is 1 but 1 is not an element of the set $a(n)=1+1/n$ so it is not a minimum. Same for maximum and supremum.
