limit of sequence as sum of series

If $U$ is the limit of a sequence

$$u_1,u_2,u_3,\mathrm{\dots }$$

of real or complex numbers, then $U$ can be expressed as the series sum

$$U=u_1+\sum _{i=1}^{\mathrm{\infty }}(u_{i+1}-u_i).$$

Proof. Let  $s_n:=u_1+{\displaystyle \sum _{i=1}^{n-1}}(u_{i+1}-u_i)$.  We see that

$$s_n=u_1+\sum _{i=1}^{n-1}{u}_{i+1}-\sum _{i=1}^{n-1}{u}_i=u_1+\sum _{j=2}^n{u}_j-\sum _{i=1}^{n-1}{u}_i=u_n$$

for all  $n=1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\dots }$  Thus

$$u_1+\sum _{i=1}^{\mathrm{\infty }}(u_{i+1}-u_i)=\underset{n\to \mathrm{\infty }}{lim}{s}_n=\underset{n\to \mathrm{\infty }}{lim}{u}_n=U,$$

Q.E.D.