My teacher really takes account about the rigor of a proof. The following is the proposition and a proof given by myself. Can someone please check it whether it is rigorous enough?

Proposition: Let be a sequence that converges to . Show that if for all , then .

Proof: Let , then there exists such that implies . Now take and implies

Since , we must have

Since the difference can be taken arbitrarily small, or even negative. This proves that .

I really doubt the rigor of this proof. And after I type in the proof, I doubt the validity.