Firstly, my lecturer defined this subject as, There is a set S, and the elements inside this set is the limits of the subsequences of that sequence, and he defined limsupxn as the supremum of the set S. In other words, the largest limit of all subsequences.

He gave this example: {xn}=1/n. and wrote liminfxn=0<1=limsupxn

1/n converges to zero , and therefore all of his subsequences will converge to zero. then why is 1=limsupxn ? What am I missing.

and my second question is, why did they call it limsupxn and not limmaxxn? This is a stupid question I know but isn't every element of S is a a limit of one of the subsequences of xn itself? so it belongs to the set S after all isn't it? so it's actually the maximum of the set S. So limsupxn is just a name or is there a situation that a limit doesn't belong to the set S but it's the supremum and we can still count it as a limit f a subsequence?

I hope my questions were clear enough. Thank you.