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 n, let p_{n} = 1 - \frac{1}{n}. Show that the sequence  p_{1}, p_{2}, p_{3}, ... converges to 1.

Proof:
To show that p converges to 1, we must show that if S is an open interval containing 1, then \exists a positive integer N such that if n is a positive integer and n >= N, then p_{n} \in S. Let S be an open interval containing 1 and let n \in \mathbb{Z}^+ and N \in \mathbb{Z}^+. Let S be the interval (1-\epsilon, 1+\epsilon), \epsilon > 0. Then |p_{n} - 1| < \epsilon in order for p_{n} \in S. Since p_{n} = 1 - \frac{1}{n}, |p_{n} - 1| = |1 - \frac{1}{n} - 1| = |-\frac{1}{n}| = |-1 * \frac{1}{n}| = |\frac{1}{n}|. And |\frac{1}{n}| < \epsilon. Since n is positive, we can say \frac{1}{n} < \epsilon \Longrightarrow \frac{1}{\epsilon} < n. Now let N = \frac{1}{\epsilon}. Since for all n > N p_{n} \in S, p converges to 1.

A few things I'm concerned about are at the end when I let N = \frac{1}{\epsilon}; if I say this is it implied that \frac{1}{\epsilon} is a positive integer, or can that not be assumed because we don't know that 1 evenly divides \epsilon? Also, I showed for n > N instead of n >= N but because of the way I formed the interval I'm not sure how to fix that. Any tips would be greatly appreciated!