0
$\begingroup$

I know that limit inferior describes the smallest number, v, that a subsequence can converge to, and I know that limit superior describes the largest number, u, that a subsequence can converge to.

So if lim inf = lim sup, then all subsequences must converge to the same number, and if all convergent subsequences converge to a limit, L, then the sequence itself converges to a limit L. Is this correct thinking?

For the converse, would I say that if a sequence converges to a limit, L, then any subsequence converges to L, and by proxy, if all subsequences converge to L, lim sup = lim inf?

$\endgroup$
0
$\begingroup$

Looks good. For the other direction, you might bring into play the fact that there are subsequences of $\{x_n\}$ that converge to $\limsup x_n$ and $\liminf x_n$ respectively. You also don't need the boundedness hypothesis if you define convergence of a sequence $\{x_n\}$ to $\pm\infty$ in the natural way.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.