# Given $\lim _{n\to \infty} a_n = L$ ($L>0$), how to prove that $\lim_{n \to \infty} 1/a_n = 1/L$ without knowing the expression of $a_n$? [duplicate]

Given positive sequence $$a_n$$ where $$\lim _{n\to \infty} a_n = L, L >0$$, prove using the limit definition that $$\displaystyle{\lim _{n\to \infty}}\frac{1}{a_n} = \frac{1}{L}.$$

My thoughts:

How can I do this if I don't know how $$a_n$$ is defined? I can use the given limit to get the range of $$a_n$$ in terms of $$L$$, but I lack the direction to complete the proof.

You don't know the expression of $$a_n$$ though you know the limit of the sequence as $$n\to\infty$$ is $$L$$, which gives you useful information:
• (1) the sequence $$(a_n)$$ is bounded;
• (2) $$|a_n-L|$$ is small when $$n$$ is large;
• (3) $$a_n$$ is away from $$0$$ when $$n$$ is large since $$L>0$$.
Note that $$\left|\frac{1}{a_n}-\frac{1}{L}\right|=\frac{|a_n-L|}{|a_n|\cdot L}.$$
You want to show that this quantity is small for large $$n$$. Well, you can make the numerator small by (2). Now use (3) to show that $$1/|a_n|$$ cannot be too big.