I'm studying for an exam and I'm not too sure how to do this problem.
Let $a_n$ be a sequence in $\mathbb R$ with $\lim_{n \rightarrow \infty}a_n = -1$. Prove that there exists $n_0 \in \mathbb N$ such that $a_n < -\frac{1}{2}$ for all $n \ge n_0$
I'm studying for an exam and I'm not too sure how to do this problem.
Let $a_n$ be a sequence in $\mathbb R$ with $\lim_{n \rightarrow \infty}a_n = -1$. Prove that there exists $n_0 \in \mathbb N$ such that $a_n < -\frac{1}{2}$ for all $n \ge n_0$
$a_n\to -1<=>\forall ε>0$ there is a $n_0:|a_n+1|<ε$ for a every $n\geq n_0$.This means that $-1-ε<a_n<ε-1$ for every $n\geq n_0$. Take $ε=\frac {1}{2}$.