The limsup (limit superior) of a sequence is the largest limit point of the sequence, while the lim (limit) is the value that the sequence approaches as the number of terms in the sequence approaches infinity.
To prove that the limsup of a sequence is equal to the lim of the sequence, you need to show that the limsup is both an upper bound and a limit point of the sequence. This can be done by using the definition of the limsup and lim, along with the properties of limit points and upper bounds.
Yes, the limsup and lim of a sequence can be different. This can happen when the sequence has multiple limit points or when the sequence does not converge to a single value.
Proving that the limsup of a sequence is equal to the lim of the sequence helps to establish the convergence of the sequence. It also allows us to evaluate the behavior of the sequence as the number of terms increases, providing insights into the long-term behavior of the sequence.
Yes, some common techniques for proving the limsup of a sequence is equal to the lim of the sequence include using the definition of limsup and lim, using the properties of limit points and upper bounds, and using the squeeze theorem.