To show point-wise convergence, we must show that the sequence $\{f_n(x)=\sum_{i=1}^{n} c_i I(x-x_i)\}$ converges to $f(x)=\sum_{i=1}^{\infty}c_iI(x-x_i)$ for every value of $x$. This will follow trivially from the definition of an infinite series if we can show that the sequence converges at all. Since the space of real numbers is complete, it suffices to show that the sequence $\{f_n(x)=\sum_{i=1}^{n} c_i I(x-x_i)\}$ is Cauchy, that is for each $e > 0$ we can find an $N$ such that $\sum_{i=N}^{\infty} c_i I(x-x_i) < e$. But we can certainly find such an $N$ for $\sum_{i=N}^{\infty} c_i > \sum_{i=N}^{\infty} c_i I(x-x_i)$, thus we have that the sequence is point-wise convergent.
Uniform convergence, on the other hand, requires us to be able to choose $N$ independently of the value of $x$. However, we did not use the value of $x$ in selecting $N$ above, so uniform convergence follows as well.