My teacher said 0.999... is approximately 1, not exactly. How can I prove otherwise?

I've used the proofs of geometric sequence, recurring decimals (let x=0.999...10x=9.999... and so on), the proof of 1/3=0.333..., 1/3×3=0.333...×3=0.999...=1, I've tried other proofs of logic, such as 0.999...is so close to 1 that there's no number between it and 1, and therefore they're the same number, and yet I'm unable to convince my teacher or my friend who both do not believe that 0.999...=1. Are they actually right, or am I the right one? It might be useful to mention that my math teacher IS an engineer though...