Introduction to Mathematical Thinking-Problem 8

8.Prove that if the sequence { a n } n = 1 ∞ \left\{a_n\right\}_{n=1}^\infty {an​}n=1∞​ tends to limit L as $n\to \infty$, then for any fixed number $M>0$, the sequence { M a n } n = 1 ∞ \left\{M a_n\right\}_{n=1}^\infty {Man​}n=1∞​ tends to the limit $ML$.

Proof: Because the sequence { a n } n = 1 ∞ \left\{a_n\right\}_{n=1}^\infty {an​}n=1∞​ tends to limit L as $n\to \infty$, for any $ϵ$, we can find an $m$ such that for all $n⩾m$, we have $|a_n-L|⩽ϵ$.
Multiply both sides by $M$, so we have $M|a_n-L|⩽Mϵ$. By algebra, we have$|M{a}_n-ML|⩽Mϵ$, for any $Mϵ$.Therefore, the sequence { M a n } n = 1 ∞ \left\{M a_n\right\}_{n=1}^\infty {Man​}n=1∞​ tends to the limit $ML$.