Prove that the sequence $\{a_n|a_n \geq 6 \}$ has finitely many elements

Question

Suppose the sequence $\{a_n\}$ converges to $5$ then prove that the sequence $\{a_n|a_n \geq 6 \}$ has finitely many elements.

I don't understand how to prove this using the fact that the sequence converges.

Answer 1

Guide:

If $a_n$ converges to $5$, then for any $\epsilon > 0$, there exists $N_\epsilon >0,$ such that $n>N_\epsilon$, then $|a_n-5|< \epsilon$.

Choose your $\epsilon$ wisely.

Answer 2

More guide: There is an $N$ such that for all $n\geq N$, we have $|a_{n}-5|<1$, then $a_{n}<5+1=6$ for all such $n$.