why its not true??

in each of the following sequence our temporary inf

gets bigger or stays the same

Correct.
so the function of the inf's goes up

Correct.
and their limit is the least upper bound(b)

It is the least upper bound of that sequence of temporary infs. It is not the least upper bound of the original sequence.
and the least upper bound is bigger or equal than every member

in the sequence.

So the lim inf is greater than or equal to each of the numbers . That does not make it greater than or equal to a_n.
can you tell me whats wrong with what i said?