# Math Help - Subsequential limits

1. ## Subsequential limits

Find the set S of subsequential limits, the limit superior, and the limit inferior.

(tn)=(0,1,2,0,1,3,0,1,4,....)

2. 0 and 1 seem to me to be subsequential limits. 1 is greater than 0. No?

3. Originally Posted by noles2188
Find the set S of subsequential limits, the limit superior, and the limit inferior.

(tn)=(0,1,2,0,1,3,0,1,4,....)
Any convergent subsequence much have its terms sufficiently close to one another (Cauchy sequence). However, all these terms are composed of integers so the only way to get a convergent subsequence is that it is eventually constant. Thus, convergent subsequences must eventually be composed of entirely 0's or entirely 1's. Thus, 0 and 1 are the subsequential limits. Therefore, the limit superior is 1 while the limit inferior is 0.