nonzero sequence of numbers with arbitrarily small sum

Question

Is it possible to have sequence of numbers $\{f(a)\}$ with the following properties:
$\sum_{i=1}^{\infty} f_i(a)<a$, for all a>0
also, $f_i(a)>0$
can someone give a sequence with these properties.

Answer 1

Let $g_i(a):=\frac{f_i(a)}a>0$. It suffices to have $$\sum_i g_i(a)<1$$ and this is can be achieved by any convergent positive series (with a scaling factor), in particular with a definition independent of $a$.

Answer 2

$$ \sum _{n=2}^{\infty} \frac {a}{2^n} = \frac {a}{2} <a $$