This is quoted from wikipedia,
The limit superior of $x_n$ is the smallest real number $b$ such that, for any positive real number $\epsilon$ , there exists a natural number $N$ such that $x_n < b + \epsilon$ for all n > N . In other words, any number larger than the limit superior is an eventual upper bound for the sequence. Only a finite number of elements of the sequence are greater than $b + \epsilon$ .
Now doesn't all the elements lie within $b$ i.e. $\forall x \in <x_n> , x \leq b $ ? Isn't the limit superior the upper bound of the sequence? If so, why should there be any elements greater than $b$ as said in the bolded statement of wikipedia above? How can there be elements, though finite, greater than $b$? Where am I mistaking? Please help as I am new in this topic. Thanks in advance.