Prove a sequence converges to 1
I am new to the world of proofs and I'm working on an assignment for Analysis. While I understand the concept of convergence I'm finding that writing the proofs is difficult for me and I want to ask if my logic is sound here and what I can improve on.
Problem: For each positive integer , let . Show that the sequence converges to 1.
To show that converges to 1, we must show that if is an open interval containing 1, then a positive integer such that if is a positive integer and , then . Let be an open interval containing 1 and let and . Let be the interval , . Then in order for . Since , . And . Since is positive, we can say . Now let . Since for all , converges to 1.
A few things I'm concerned about are at the end when I let ; if I say this is it implied that is a positive integer, or can that not be assumed because we don't know that 1 evenly divides ? Also, I showed for instead of but because of the way I formed the interval I'm not sure how to fix that. Any tips would be greatly appreciated!