# Thread: Prove that there exists a positive integer n

1. ## Prove that there exists a positive integer n

1. Suppose delta > 0.

(a) Prove that there exists a positive integer n such that

0 < 1/((4n+1)pi/2) < delta

and sin ((4n+1)pi/2) = 1

(b) Prove that there exists a positive integer m such that

0 < 1/((4m+3)pi/2) < delta

and sin ((4m+3)pi/2) = -1

see if anyone can do it,

2. Originally Posted by 450081592
1. Suppose delta > 0.

(a) Prove that there exists a positive integer n such that

0 < 1/((4n+1)pi/2) < delta

and sin ((4n+1)pi/2) = 1

(b) Prove that there exists a positive integer m such that

0 < 1/((4m+3)pi/2) < delta

and sin ((4m+3)pi/2) = -1

see if anyone can do it,
For the first one take $n> \frac{1}{4} (\frac{1}{ \delta \pi /2} -1 )$ (just isolate $n$) and note that $\sin ((4n+1) \pi /2)=\sin (2n\pi ) \cos (\pi /2) + \sin (\pi /2) \cos (2n \pi )=\sin (\pi /2)=1$. For the second one do exactly the same thing, the only difference being $\sin (3\pi /2)=-1$

3. what does this imply? I cant see the connection between this and n to be an positive integer, can you explain it please