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 $a_n$?

Question

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 as $n\to\infty$ is $L$. — , Apr 02 '19 at 21:25
That's a great question..and proving this theorem will give you a bit of insight: sometimes you don't need to know a "formula" for something to be able to prove something useful about it. Think of the chain rule: you don't know the formulas for $f$ and $g$, but you know that if they're differentiable at the right points, then $(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$, which is pretty amazing! — John Hughes, Apr 02 '19 at 21:29
Possible duplicate of How to formally prove that if $\lim \limits_{n \to \infty}a_n=\infty$, then $\lim \limits_{n \to \infty}\frac{1}{a_n}=0$.. Also, see https://math.stackexchange.com/questions/1171733/convergence-of-inverse-of-convergent-sequence/1171755#1171755 and https://math.stackexchange.com/questions/54754/proving-that-b-n-to-b-implies-left-frac1b-n-right-to-frac1b (A better duplicating target) — YuiTo Cheng, Jul 08 '19 at 12:36

score 1 · Accepted Answer · 2019-04-02T21:37:32.773

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.

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 $a_n$?

1 Answers1