There are many misconceptions in your question and in your proposed solution. I won't solve the problem for you, but I'll explain the problems in your solution and I'll give you a hint how to find the correct answer.
First of all, I suppose that $u[n]$ is the unit step function, and consequently
The confusion about multiplication and convolution has already been pointed out by Dilip Sarwate in his comment.
Now for the correct way to solve this problem:
If $a[n]$ is the system's step response, i.e. $a[n]=(h*u)[n]$, then, due to linearity and time-invariance, we can write the response $y[n]$ to the given input $x[n]=u[n]-u[n-1]$ as
So the only thing you need to know is the system's step response $a[n]$, which is obtained by convolving $h[n]$ with $u[n]$. If you write down the convolution sum then you should find that in this case
With the given impulse response $h[n]$, evaluating (2) is simple because it is a geometric series. As soon as you have $a[n]$, the output is directly obtained from (1).
A much simpler route is to realize that the input $x[n]=u[n]-u[n-1]$ is equal to a unit impulse $x[n]=\delta[n]$. This means that the output $y[n]$ is simply given by the impulse response: $y[n]=h[n]$. Note that if you do things right, then this solution and the general solution given by (1) are identical.