Good night. I make a proof of this prove:

Prove $\bigcap\limits_{n=1}^{\infty}(-\frac{1}{n},1+\frac{1}{n})=\left[0,1\right]$


Be $\left\{ x_{n}\right\} =\left\{ -\frac{1\;}{x}:\;x\;\epsilon\:\mathbb{N}\right\} $ and $\left\{ Y_{n}\right\} =\left\{ \frac{1}{y}+1\;:\;y\;\epsilon\:\mathbb{N}\right\}$ sequences such then $lim_{n\rightarrow\infty}\frac{1}{x_{n}}=0$ and $lim_{n\rightarrow\infty}\frac{1}{y_{n}}+1=1$ but when i see the intersection that confused me because the intersection is $Ø$. Can someone help me?


Hint 1: $[0,1] \subset(-\frac{1}{n},1+\frac{1}{n})$.

Hint 2: If $x \in \cap_{n=1}^{\infty}(-\frac{1}{n},1+\frac{1}{n})$ then $$-\frac{1}{n} < x < 1+\frac{1}{n}$$

What happens when you take the limit by $n$?

  • 2
    $\begingroup$ I'm an idiot. sorry -.- Good answer! $\endgroup$ – Bvss12 Aug 3 '16 at 2:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.