I've been trying to prove (maybe even disprove) the following inequality: $$ \sum_{n=1}^{N} \frac{a_n}{\sqrt{\sum_{i=1}^{n}a_i}} \leq C \sqrt{\sum_{n=1}^{N}a_n} $$ Where $ a_1,...,a_N\geq 0 $ are some nonnegative numbers, and $C$ is an absolute constant. Help will be much appreciated.

4$\begingroup$ $C$ is absolute means that it is independent of $N$, right? Also, although it probably doesn't much matter, you say symbolically that the $a$'s are nonnegative, but then write that they are positive. $\endgroup$ – LSpice Apr 12 '20 at 23:45

1$\begingroup$ @LSpice You're right, my bad. As Iosif Pinelis expalined though, it doesn't really matter $\endgroup$ – GuyK Apr 13 '20 at 10:25
For every $n\in\{1,\dotsc,N\}$, we have $$2\sqrt{\sum_{i\leq n} a_i}2\sqrt{\sum_{i\leq n1} a_i}=\frac{2a_n}{\sqrt{\sum_{i\leq n} a_i}+\sqrt{\sum_{i\leq n1} a_i}}>\frac{a_n}{\sqrt{\sum_{i\leq n} a_i}}.$$ Summing these up, we obtain the inequality with $C=2$. It is also straightforward to see that for $C<2$ the inequality fails, hence $C=2$ is the optimal constant.
Rewrite your inequality as $$lhs:=\sum_{n=1}^N \frac{s_ns_{n1}}{\sqrt{s_n}}\le C\sqrt{s_N},$$ where $s_n:=\sum_{i=1}^n a_i$. Note that $$\sum_{n=1}^N \frac{s_ns_{n1}}{\sqrt{s_n}}$$ is a lower Riemann sum for the integral $$\int_0^{s_N}\frac{ds}{\sqrt s}=2\sqrt{s_N}.$$ So, $$lhs\le2\sqrt{s_N},$$ as desired.
In the above proof, it was tacitly assumed that $a_i>0$ for all $i$. This can be obviously extended to the case when we only know that $a_i\ge0$ for all $i$  assuming that, by continuity, $\frac{a_n}{\sqrt{\sum_{i=1}^{n}a_i}}:=0$ whenever $a_n=0$.