1. ## Proof lim 1/a_n=inf

Hellou

I'm havin real troubles with this one:

Proof that if $lim_{n->\infty} a_n=0$ then $lim_{n->\infty} 1/a_n=+\infty
$

can anyone help me?

thx

2. Originally Posted by hiddy
Hellou

I'm havin real troubles with this one:

Proof that if $lim_{n->\infty} a_n=0$ then $lim_{n->\infty} 1/a_n=+\infty
$

can anyone help me?

thx
We must assume, of course, that $\{a_n\}$ is either positive or positive almost always, otherwise the result is false (for example, $\frac{-1}{n}\xrightarrow [n\to \infty]{}0$ and $-n\xrightarrow [n\to \infty]{}-\infty$ , or take any sequence which converges to zero and has infinite number of zeroes among its elements... )

Let us take any $0 (we can assume positiviness since the sequence is positive).
No , since $a_n\xrightarrow [n\to \infty]{}0$ we get that for $\epsilon=\frac{1}{R}$ there exists $N_\epsilon\in\mathbb{N}\,\,\,s.t.\,\,\,n>N_\epsilo n\Longrightarrow a_n<\frac{1}{R}$ $\Longrightarrow \frac{1}{a_n}>R\,\,\,\forall\,n>N_\epsilon$ and this means, according to the definition, that $\frac{1}{a_n}\xrightarrow [n\to \infty] {}\infty$

Tonio

3. thx!!!

ps:
i forgot to mention that all a_n are positive real numbers!