The discrete time unit step is given by $$ u[n] = \begin{cases} 1, & n \geq 0, \\ 0, & n < 0. \end{cases} \tag{1} $$
Unit Sample sequence or Impulse sequence is defined as $$ δ[n] = \begin{cases} 1, & n = 0, \\ 0, & n ≠ 0. \end{cases} \tag{2} $$
and they are related via the sum relation by $$ u[n] = \sum_{k = -\infty}^\infty \delta[k]. \tag{3}$$
The Question is : how to verify the relation between these ? $$ u[n] = \sum_{m = 0}^\infty \delta[n-m] = \sum_{k = -\infty}^\infty \delta(k), $$ $$\delta[n]=u[n]-u[n-1].\tag{4} $$
Thank you very much.