Question

Let {an}

∞

n=1 be a convergent sequence with limit L. Then use the definition of

limit to prove that

lim

n→∞

−an = −L.

Now if L = 0, then use the definition of limit to prove that

limn→∞

(−1)n

an = 0

Answer 1

Let $\{a_n\}_{n=1}^{\infty}$ be a convergent sequence with limit $L$. By the definition of limit for any $\epsilon>0$ there exists $n_0\in\mathbb N$ such that $|a_n-L|<\epsilon$ for any $n\ge n_0$. Since $|(-a_n)-(-L)|=|-(a_n-L)|=|-1|\cdot|(a_n-L)|=|a_n-L|$, we conclude that for any $\epsilon>0$ there exists $n_0\in\mathbb N$ such that $|(-a_n)-(-L)|=|a_n-L|<\epsilon$ for any $n\ge n_0$. Therefore, $\lim\limits_{n\to\infty}(-a_n)=-L.$

Now if $L=0$, then for any $\epsilon>0$ there exists $n_0\in\mathbb N$ such that $|(-1)^na_n|=|(-1)^n|\cdot|a_n|=|a_n|=|a_n-0|<\epsilon$ for any $n\ge n_0$. Therefore, by the definition of limit we have that $\lim\limits_{n\to\infty}((-1)^na_n)=0.$