Determining if a system is Linear
To show that a system

$$y[n] = T\{x[n]\} $$

is linear, you have to show that it's additive and homogeneous which is codified in

$$T\{ a ~x_1[n] + b ~ x_2[n] \} = a ~T\{x_1[n]\} + b ~T\{x_2[n] \} $$ for all complex $a,b$ and all $x_1,x_2$. 

The property set above is satisfied for a system given by :
$$y[n] = T\{x[n]\} = \sum_{k=-\infty}^{\infty} h[k] x[n-k] = h[n] \star x[n] $$ where $\star$ is the convolution operator and $h[n]$ is some sequence to be defined (incidentally it will be the impulse response of the given LTI system). 

Of course you didn't have to prove it, as it's already known that if the input-output relation of a system is given by convolution sum; i.e. $y[n] = h[n] \star x[n]$, then it's by definition an LTI (linear and time-invariant) system. Indeed input output relation of any LTI system is given by the convolution sum...