Introduction to Mathematical Thinking-Problem 8

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

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

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