$\displaystyle
\liminf x_n \leq \limsup x_n \leq \limsup X_n = \lim X_n
$
why
$\displaystyle
limsupx_n \leq \limsup X_n
$
X_n is a subsequence which converges
the definition of limsup
is being the sup of the limit group of every subsequence
you cant do limsup X_n
by doing that you split this subsequence into a smaller convergent subsequences
and take the Sup of their limits
which has nothing to do with limsup x_n
If $\displaystyle A,B$ are non-empty bounded subsets with $\displaystyle A\subseteq B$ then $\displaystyle \sup (B) \leq \sup (A)$.
Now, $\displaystyle \{ x_{r_k} | k\geq n\} \subseteq \{ x_k | k\geq n\}$.
Therefore, $\displaystyle \sup \{ x_k | k\geq n\} \leq \sup \{ x_{r_k} | k\geq n\}$.
Thus, $\displaystyle \lim \sup \{x_k |k\geq n\} \leq \lim \sup \{ x_{r_k}| k\geq n\}$.
This gives, $\displaystyle \limsup x_n \leq \limsup X_n$