Find all the real number sequences $\{b_n\}_{n \geq 1}$ and $\{c_n\}_{n \geq 1}$ that satisfy the following conditions:
(i) For any positive integer $n$, $b_n \leq c_n$;
(ii) For any positive integer $n$, $b_{n+1}$ and $c_{n+1}$ is the two roots of the equation $x^2+b_nx+c_n=0$.
To find all the real number sequences $\{b_n\}_{n \geq 1}$ and $\{c_n\}_{n \geq 1}$ that satisfy the given conditions, we will proceed step-by-step.

1. **Initial Conditions and Roots**:
   - Given that for any positive integer $n$, $b_n \leq c_n$.
   - For any positive integer $n$, $b_{n+1}$ and $c_{n+1}$ are the two roots of the quadratic equation $x^2 + b_n x + c_n = 0$.

2. **Quadratic Equation Roots**:
   - The roots of the quadratic equation $x^2 + b_n x + c_n = 0$ are given by:
     \[
     x = \frac{-b_n \pm \sqrt{b_n^2 - 4c_n}}{2}
     \]
   - Let $b_{n+1}$ and $c_{n+1}$ be the roots, then:
     \[
     b_{n+1} = \frac{-b_n + \sqrt{b_n^2 - 4c_n}}{2}, \quad c_{n+1} = \frac{-b_n - \sqrt{b_n^2 - 4c_n}}{2}
     \]

3. **Case Analysis**:
   - Suppose $b_1 \neq 0$ and $c_1 \neq 0$. We need to consider the signs of $b_n$ and $c_n$.

4. **Sign Analysis**:
   - If $b_n$ and $c_n$ are both positive or both negative, we need to check the discriminant $b_n^2 - 4c_n$ to ensure the roots are real.
   - If $b_n$ and $c_n$ are of opposite signs, we need to ensure $b_n \leq c_n$ holds.

5. **Contradiction**:
   - For every case, there exists an $n$ such that $b_n, c_n < 0$. Then:
     \[
     b_{n+1}^2 - 4c_{n+1} = \left(\frac{-b_n - \sqrt{b_n^2 - 4c_n}}{2}\right)^2 - 4 \left(\frac{-b_n + \sqrt{b_n^2 - 4c_n}}{2}\right)
     \]
     Simplifying this expression:
     \[
     b_{n+1}^2 - 4c_{n+1} = \left(\frac{b_n^2}{2} + 2b_n - c_n\right) - \left(\frac{-b_n}{2} + 2\right) \sqrt{b_n^2 - 4c_n}
     \]
     This expression simplifies to:
     \[
     b_{n+1}^2 - 4c_{n+1} < \left(\frac{b_n^2}{2} + b_n\right) - \left(\frac{-b_n}{2} + 2\right)(-b_n) = 3b_n < 0
     \]
     Hence, $x^2 + b_{n+1}x + c_{n+1} = 0$ does not have a real root, which is a contradiction.

6. **Special Cases**:
   - If $b_1 = 0$, then $c_1 \geq 0$ but at the same time $c_1 \leq 0$ for $x^2 + b_1 x + c_1 = 0$ to have a real root. Hence, $c_1 = 0$.
   - If $c_1 = 0$, then $b_2 = 0$, $c_2 = -b_1$. By a similar argument as above, we get $c_2 = 0$. Hence, $b_1 = 0$.

Therefore, the only sequences that satisfy the given conditions are $b_n \equiv 0$ and $c_n \equiv 0$.

The final answer is $\boxed{b_n \equiv 0, \; c_n \equiv 0}$.