To prove that any integer $n \geq 1$ can be expressed in the form

$n = \sum_{k=1}^{m} c_k \cdot k!$

where each $c_k$ is an integer satisfying $0 \leq c_k \leq k$, we will use mathematical induction and apply the division algorithm.

Step 1: Base Case

Let's start with $n = 1$. We want to express $1$ in the required form.

$1 = 1 \cdot 1!$

Here, $m = 1$, and we have $c_1 = 1$, which satisfies $0 \leq c_1 \leq 1$. So the base case holds.

Step 2: Inductive Hypothesis

Assume that for some integer $n \geq 1$, we can express $n$ in the form

$n = \sum_{k=1}^{m} c_k \cdot k!$

where $0 \leq c_k \leq k$ for each $k$ from $1$ to $m$.

Step 3: Inductive Step

We need to show that $n + 1$ can also be written in this form.

1. Find Suitable $m$: Let $m$ be the largest integer such that $m! \leq n + 1$.

2. Apply the Division Algorithm to $n + 1$: Using the division algorithm, divide $n + 1$ by $m!$:

   $n + 1 = q \cdot m! + r$

   where $0 \leq r < m!$ and $q$ is an integer.

3. Analyze $q$: Since $r < m!$, $q$ can be at most $m$ (because if $q$ were greater than $m$, $q \cdot m!$ would exceed $(m+1)!$, which contradicts $m$ being the largest integer such that $m! \leq n + 1$).

4. Construct Representation for $n + 1$: Write $n + 1$ as:

   $n + 1 = q \cdot m! + r$

   where $0 \leq q \leq m$ and $0 \leq r < m!$. By the induction hypothesis, $r$ can be expressed in the form

   $r = \sum_{k=1}^{m-1} c_k \cdot k!$

   where $0 \leq c_k \leq k$ for $k = 1, 2, \ldots, m-1$.

5. Combine Terms: Substitute this expression for $r$ back into our equation for $n + 1$:

   $n + 1 = q \cdot m! + \sum_{k=1}^{m-1} c_k \cdot k!$

   This gives us an expression for $n + 1$ in terms of factorials up to $m$, and each coefficient $c_k$ satisfies $0 \leq c_k \leq k$.

Conclusion

By induction, we have shown that for every integer $n \geq 1$, $n$ can be expressed in the form

$n = \sum_{k=1}^{m} c_k \cdot k!$

where each $c_k$ is an integer satisfying $0 \leq c_k \leq k$.

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Would you like further clarification or have questions on specific parts? Here are some follow-up questions you might find helpful:

1. How can we extend this proof to cover specific examples with larger $n$?
2. Can this representation be unique for each $n$? Why or why not?
3. How does the constraint $0 \leq c_k \leq k$ affect the possible values for $c_k$?
4. How does the division algorithm specifically ensure that each $c_k$ remains within the required bounds?
5. Could we apply this method to other types of number representations, like binary or hexadecimal?

Tip: Understanding how the division algorithm works in modular contexts can help with proofs that require range constraints on coefficients.