5. Determine the smallest real number $c$ such that for any sequence satisfying the condition:
$$
x_{1}+x_{2}+\cdots+x_{n} \leqslant x_{n+1}(n=1,2, \cdots)
$$
(Let $x_{1}=2^{-1}$, satisfying the problem's conditions. We can get $c \geqslant \frac{1}{\sqrt{2}-1}$. $\frac{1-\sqrt{\frac{1}{2^{n}}}}{\sqrt{1-\frac{1}{2^{n}}}}$. Let $n \rightarrow+\infty$. We get $c \geqslant \sqrt{2}+1$. Below, we use mathematical induction to prove $\sum_{i=1}^{n} \sqrt{x_{i}} \leqslant(\sqrt{2}+1) \cdot \sqrt{\sum_{i=1}^{n} x_{1}}$. Thus, we get $c_{\text {min }}$ $=\sqrt{2}+1$.)