The reason why is because is convergent.
The rest of them are similar.
this expression is not correct
X_n is bounded
the limit of the sequence doesnt equal the limsup
Xn is not convergent
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