Michael - yes, that's good! Great!
You would need some feedback, and practice to learn how to write up proofs the way mathematicians do. This is math grad students spend so much time doing homeworks full of proofs, and I spend so much time grading these, writing comments in the margin, and showing all the students the 'best' proofs, so they can learn what good examples are like.
However, you didn't sign up for math grad school, and I'm not being paid to teach this course, so I won't correct the style of your proof. I will just say: you've got the idea right. :)>-