electromagnetism - Proof that four-potential is a four-vector

My teacher proposed this "simple" proof that the 4-potential is a 4-vector which I am very skeptic about.

Since under gauge transformation the 4-four potential transforms as

$$A^\mu \mapsto A^\mu + \partial^\mu\lambda,$$ $\lambda$ being a scalar function, it follows that $A^\mu$ must trasform as a 4-vector under Lorentz transformation, since $\partial^\mu$ is one.

Is he right? What am I missing? I asked him for clarification but didn't get any more information other than this.