I'm trying to proof a basic statement: if $|a| &lt; 1$, $a\in \mathbb{R}$ then $\lim_{n\rightarrow \infty } a^n = 0$. On one hand, it's quite intuitive, on the other hand, I wasn't able to prove it formally. Is this statement needs to be proved or it's some kind of convention? Thank you.
Given $|z|&lt;1$ with $z \in \mathbb{C}$ prove $\lim_{n \to \infty} z^n=0$.  My attempt given $\epsilon &gt;0$ we wish to show that there exists $\delta &gt;0$ such that $n&gt; \delta$ means $|z^n-0| &lt; \epsilon$.  For $n \in \mathbb{N}$. We have $|z^n|=|z|^n$. We wish $|z|^n &lt; \epsilon$. This means,  $$n \ln |z| &lt; \ln \epsilon$$  But because $\ln |z|&lt;\ln 1=0$ dividing both sides by $\ln |z|$ gives:  $n&gt; \frac{\ln \epsilon}{\ln |z|}$  So that's what we choose for $\delta$.   My question is is how can we get away from the assumption that $n \in \mathbb{N}$. Or is this correct even with the assumption?
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