The system is defiantly not time invariant. When the signal is shifted the output is shifted also but a different magnitude is applied to the output, showing that time affects the output.
Sine we are talking about DT a setup of
$ X[n]=\sum_{k=-\infty}^{\infty} $ δ[n-k].
However for $ u[n] $ we will need to cap off the $ -\infty $ to 0, since the for all values less then 0 the unit step would produce 0.
X[n]=$ \sum_{k=0}^{\infty} $ δ[n-(k+1)].
