I am wondering, $lim_{n\to\infty} a_n =L$, then ${a_n}$ is bounded both-side, is not a perfect statement.
can we say that ${a_n}$ is bounded above? or it is unsure it is bounded above or below?
Does there exist a sequence where $lim_{n\to\infty} a_n =L$, but $lim_{n\to-\infty}a_n$ does not exist?
1
$\begingroup$
real-analysis
limits
convergence
-
0Sure, why not ? – 2017-02-10
-
0can we say that $a_n$ is bounded above? or it is unsure it is bounded above or below? – 2017-02-10
-
4you can just define $a_n:n\ge 0$ as any convergent sequence and $a_n:n\le 0$ as any divergent squence; they are just separate sequences where you index them as though they are a single sequence. And $a_n$ might not be bounded above depending on its behavior as $n\to -\infty$ – 2017-02-10
1 Answers
6
$(-1)^ne^{-n}$ ..................