Consequences of $\limsup\limits_{n \rightarrow \infty}(a_n)=1$.

Question

$\{{a_n}\}$ is a sequence with $\limsup\limits_{n \rightarrow \infty}(a_n)=1$.

I have trouble with these, so I want to verify these

(a) $a_n \leq1$ eventually; False

(b) $a_n \leq2$ eventually; True

(c) $a_n \geq1$ eventually; False

If any of these are wrong, could you please provide me with a counter-example so that I can verify it for myself? I'm afraid the "eventually" part is unclear to me.

Edit:

I actually intended for (c) to read $a_n \geq0$ eventually; False

Answer 1

For (a), which is indeed false, a counterexample is $a_n=1+1/n$.

For (c), which is indeed false, a counterexample is $a_n=1-1/n$.