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I am trying to complete the following question:

Suppose that $\lim_{n \to \infty} a_{n}=L$ and $L \neq 0$. Prove there is some $N$ such that $a_{n} \neq 0$ for all $n \geq N.$

Intuitively it makes sense. If after a certain point the numbers in the sequence $a_n$ can be made arbitrarily close to $L \neq 0$, then we should be able to find an $N$ for which $a_n \neq 0$ for $n \geq N.$

Let $L$ be represented by an infinite decimal expansion, then

$$L = x_{0}.x_{1}x_{2}x_{3}x_{4}x_{5}x_{6} \cdots $$

where for all $i \geq 1$, $x_i \in \left \{ 0,1,...,9 \right \}$, and $i \in \mathbb{I}$. Furthermore, because $L \neq 0$, there must be at least one $x_i \neq 0$. Now, let the first nonzero $x_i$ occur in the $k^{th}$ decimal place.

I know that $a_n \to L$ as $n \to \infty$. As such, for every $\epsilon > 0$, there is an integer $N = N(\epsilon)$ such that

$$|a_n - L| < \epsilon$$

for all $n \geq N$. Let $\epsilon = \frac{1}{10^{k+1}}$, then we have

$$L - \epsilon < a_n < L + \epsilon$$

$$x_{0}.x_{1}x_{2}x_{3} \cdots x_{k} \cdots - \frac{1}{10^{k+1}} < a_n < x_{0}.x_{1}x_{2}x_{3} \cdots x_{k} \cdots + \frac{1}{10^{k+1}}$$

$$(0.000 \cdots x_{k} \cdots) - \underbrace{0.000...0}_\text{k+1}1 < a_n < (0.000 \cdots x_{k} \cdots) + \underbrace{0.000...0}_\text{k+1}1$$

This should show that $a_n$ can be made to fall within an interval that does not contain zero for $n \geq N$. Does my reasoning make sense?

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2 Answers 2

up vote 2 down vote accepted

Yes your proof is correct.
Yet there is an easier way:
pick $\epsilon=\frac{|L|}{2}>0$ (actually any $0<\epsilon<|L|$ will do, as the one in your proof.).

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You can make this argument a lot more simple by just saying that, by definition of convergence, you can choose $\epsilon = d(L,0)$, where d is the metric on the appropriate space. Then, by convergence, $\exists N \in \mathbb{N}$ s.t. $d(L,a_n) < \epsilon$ for all $n \geq N$. Then, for these $a_n$, you'll see that they cannot be equal to zero.

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