(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that: ⋂_(n=1)^∞▒〖(0,1/n]=∅〗

2. Relevant equations

If tried with mathematical induction, we would need to know the procedure follows:

This would be my P(n) statement: ⋂_(n=1)^∞▒〖(0,1/n]=∅〗

now we would need to find out if P(1) is true and then assume P(k) for some k[tex]\geq[/tex]1, is true. Then we would need to prove the statement P(k+1) is also true.

3. The attempt at a solution

I tried using induction, but our professor, nor the book explained any examples that actually go to infinity, all we covered and proved with induction has been up to "n".

Thus my thinking consists of taking the semiintervals:

a) (0,1/1]∩(0,1/2]=(0,1/2]

so it continues for even smaller semiintervals, and the intersection between a bigger semi-interval with a larger semi-interval is the smaller of the two...up to the point of having

(0,1/(huge nr close to ∞)]∩(0,1/∞]=(0,1/∞]=(0,0]

The only thing I cannot explain is since this semi-interval contains zero, how can it be an empty set? So if someone could give me an idea or a hint, anything at all, I would appreciate it a lot!

emira

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# Advanced calculus proof

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