Given $a\in\mathbb{R}$ and $0, let $X_n=a^n$, $\forall n\in\mathbb{N}$.
Prove that $\lim \limits_{n\to \infty}X_n=0$ using limit definition or limits arithmetics(including the squeeze theorem if needed).
Thanks a lot.
Given $a\in\mathbb{R}$ and $0, let $X_n=a^n$, $\forall n\in\mathbb{N}$.
Prove that $\lim \limits_{n\to \infty}X_n=0$ using limit definition or limits arithmetics(including the squeeze theorem if needed).
Thanks a lot.
Note that $\dfrac{1}{a}>1$. Let $\dfrac{1}{a}=1+k$.
By induction, or by using the Binomial Theorem, we can show that $(1+k)^n \ge 1+nk$. It follows that $0 Now it should not be hard to use the $\epsilon$-$N$ definition, or Squeezing, to get the result.
Remark: One could use fancier tools. The sequence $(a^n)$ is decreasing. It is bounded below by $0$. So the sequence has a limit. Let $L$ be the limit. Then $L=\lim_{n\to\infty} a^n=\lim_{n\to\infty}a^{n+1}=a\lim_{n\to\infty} a^n=aL.$ so $L(1-a)=0$ and therefore $L=0$.