$a_n\geq b_n$ for $n>\bar{n}$ implies $\limsup_{n\rightarrow \infty}a_n\geq \limsup_{n\rightarrow \infty}b_n$

Question

Consider two sequences of real numbers $\{a_n\}_n, \{b_n\}_n$.

I know that if $a_n\geq b_n$ $\forall n$ then $\limsup_{n\rightarrow \infty}a_n\geq \limsup_{n\rightarrow \infty}b_n$.

Suppose instead with $a_n\geq b_n$ for $n>\bar{n}$. Can I still say $\limsup_{n\rightarrow \infty}a_n\geq \limsup_{n\rightarrow \infty}b_n$?

Answer 1

Yes. Removing finitely many terms from a sequence does not influence its accumulation points.