Let $\{a_n\}^{\infty}_0$ and $\{b_n\}^{\infty}_0$ be two sequences determined by the recursion formulas
\[a_{n+1} = a_n + b_n,\]
\[ b_{n+1} = 3a_n + b_n, n= 0, 1, 2, \cdots,\]
and the initial values $a_0 = b_0 = 1$. Prove that there exists a uniquely determined constant $c$ such that $n|ca_n-b_n| < 2$ for all nonnegative integers $n$.
1. **Define the sequences and initial conditions:**
   Given the sequences $\{a_n\}^{\infty}_0$ and $\{b_n\}^{\infty}_0$ determined by the recursion formulas:
   \[
   a_{n+1} = a_n + b_n,
   \]
   \[
   b_{n+1} = 3a_n + b_n,
   \]
   with initial values $a_0 = 1$ and $b_0 = 1$.

2. **Express $b_n$ in terms of $a_n$:**
   From the first recursion formula, we have:
   \[
   b_n = a_{n+1} - a_n.
   \]

3. **Substitute $b_n$ into the second recursion formula:**
   Using the expression for $b_n$, we substitute into the second recursion formula:
   \[
   b_{n+1} = 3a_n + b_n = 3a_n + (a_{n+1} - a_n) = 2a_n + a_{n+1}.
   \]

4. **Formulate a second-order linear recurrence relation:**
   From the above, we have:
   \[
   a_{n+2} - 2a_{n+1} - 2a_n = 0.
   \]
   This is a second-order linear homogeneous recurrence relation.

5. **Solve the characteristic equation:**
   The characteristic equation for the recurrence relation is:
   \[
   r^2 - 2r - 2 = 0.
   \]
   Solving this quadratic equation, we get:
   \[
   r = \frac{2 \pm \sqrt{4 + 8}}{2} = 1 \pm \sqrt{3}.
   \]

6. **General solution for $a_n$:**
   The general solution for $a_n$ is:
   \[
   a_n = A(1 + \sqrt{3})^n + B(1 - \sqrt{3})^n,
   \]
   where $A$ and $B$ are constants determined by initial conditions.

7. **Determine constants $A$ and $B$:**
   Using the initial conditions $a_0 = 1$ and $a_1 = 2$ (since $a_1 = a_0 + b_0 = 1 + 1$), we set up the system:
   \[
   A + B = 1,
   \]
   \[
   A(1 + \sqrt{3}) + B(1 - \sqrt{3}) = 2.
   \]
   Solving this system, we get:
   \[
   A = \frac{1 + \sqrt{3}}{2\sqrt{3}}, \quad B = \frac{1 - \sqrt{3}}{2\sqrt{3}}.
   \]

8. **Express $b_n$ in terms of $a_n$:**
   Using $b_n = a_{n+1} - a_n$, we find:
   \[
   b_n = A((1 + \sqrt{3})^{n+1} - (1 + \sqrt{3})^n) + B((1 - \sqrt{3})^{n+1} - (1 - \sqrt{3})^n).
   \]
   Simplifying, we get:
   \[
   b_n = A(1 + \sqrt{3})^n \sqrt{3} + B(1 - \sqrt{3})^n (-\sqrt{3}).
   \]

9. **Find the constant $c$:**
   We need to find a constant $c$ such that $n|ca_n - b_n| < 2$ for all nonnegative integers $n$. Using the expressions for $a_n$ and $b_n$, we have:
   \[
   ca_n - b_n = c\left(A(1 + \sqrt{3})^n + B(1 - \sqrt{3})^n\right) - \left(A(1 + \sqrt{3})^n \sqrt{3} + B(1 - \sqrt{3})^n (-\sqrt{3})\right).
   \]
   Simplifying, we get:
   \[
   ca_n - b_n = A(1 + \sqrt{3})^n (c - \sqrt{3}) + B(1 - \sqrt{3})^n (c + \sqrt{3}).
   \]
   For $n|ca_n - b_n| < 2$, we need $c = \sqrt{3}$, so that:
   \[
   ca_n - b_n = A(1 + \sqrt{3})^n (0) + B(1 - \sqrt{3})^n (0) = 0.
   \]

10. **Conclusion:**
    Therefore, the constant $c$ is uniquely determined as $\sqrt{3}$.

The final answer is $\boxed{\sqrt{3}}$.