Could we just say that since the set of subsequential limits is just ##\{L\}##, we have that ##\limsup a_n = \sup \{L\} = L = \inf \{L\} = \liminf a_n##. So ##\lim_{n\to\infty} a_n = L##?Math_QED said:Hint: This is easy considering ##\liminf_{n \to \infty}a_n## and ## \limsup_{n \to \infty} a_n##
Mr Davis 97 said:Could we just say that since the set of subsequential limits is just ##\{L\}##, we have that ##\limsup a_n = \sup \{L\} = L = \inf \{L\} = \liminf a_n##. So ##\lim_{n\to\infty} a_n = L##?
A subsequence is a sequence that is obtained by removing some elements from a given sequence while maintaining the relative order of the remaining elements.
If every subsequence of a sequence converges to a specific limit, then the original sequence also converges to the same limit. This is known as the "subsequence convergence theorem."
A sequence converges to a limit L if, for any small positive number ε, there exists a positive integer N such that for all n greater than or equal to N, the terms of the sequence are within ε of the limit L.
No, a sequence can only converge to one limit. If a sequence converges to more than one limit, then it is said to be a divergent sequence.
To prove that a sequence converges to a specific limit L, you must show that for any small positive number ε, there exists a positive integer N such that for all n greater than or equal to N, the terms of the sequence are within ε of the limit L. This can be done using the definition of convergence or by using the subsequence convergence theorem.