# Is this argument a valid use of inductive proof?

An inductive proof is typically done with three steps:

• Base Case
• Inductive Hypothesis
• Inductive Step

The argument that follows does not have a Base Case.

• Is this approach valid (I believe that it is -- but I could be wrong)
• Assuming this approach is valid, is my use of this approach valid (there could be a mistake in my reasoning)

Let:

$$H_n(x) = \sum\limits_{i |n\#}\left\lfloor\frac{x}{i}\right\rfloor\mu(i)$$

where:

• $$H_n(x)$$ is the count of $$t$$ where $$t \le x$$ and gcd$$(t,n\#)=1$$
• $$n\#$$ is the primorial of $$n$$
• $$\mu(i)$$ is the möbius function.

If $$p_k$$ is the $$k$$th prime, then:

$$H_{p_{k+1}}(x) = H_{p_k}(x) - H_{p_k}\left(\frac{x}{p_{k+1}}\right)$$

I was wondering if it is valid to use the following inductive argument with recurrent relations $$W_k(n), d_k(n), c_k(n)$$ where I do not define $$W_0(n), c_0(n), d_0(n)$$.

Let:

• $$W_k(x) = W_{k-1}(x) - \dfrac{W_{k-1}(x)}{p_k}$$
• $$c_k = c_{k-1} + \dfrac{d_{k-1}}{p_k}$$
• $$d_k = d_{k-1} + \dfrac{c_{k-1}}{p_k}$$
• $$U_k(x) = W_k(x) + c_k$$
• $$L_k(x) = W_k(x) - d_k$$

Rather than defining $$W_0(n), c_0, d_0$$ which may or may not exist, assume that the following is valid up to $$k$$:

• $$U_k(x) \ge H_{p_k}(x) \ge L_k(x)$$

• $$\dfrac{U_k(x)}{w} \ge H_{p_k}\left(\dfrac{x}{w}\right) \ge \dfrac{L_k(x)}{w}$$ for all $$1 \le w \le x$$

Then:

• $$H_{p_{k+1}}(x) \le U_k(x) - \dfrac{L_k(x)}{p_{k+1}} \le W_k(x) - \dfrac{W_k(x)}{p_{k+1}} + c_k + \dfrac{d_k}{p_{k+1}} = W_{k+1}(x) + c_{k+1} = U_{k+1}(x)$$
• $$H_{p_{k+1}}(x) \ge L_k(x) - \dfrac{U_k(x)}{p_{k+1}} \ge W_k(x) - \dfrac{W_k(x)}{p_{k+1}} - d_k - \dfrac{c_k}{p_{k+1}} = W_{k+1}(x) - d_{k+1} = L_{k+1}(x)$$
• $$H_{p_{k+1}}\left(\dfrac{x}{w}\right) \le \dfrac{U_k(x) - \frac{L_k(x)}{p_{k+1}}}{w} \le \dfrac{W_k(x) - \frac{W_k(x)}{p_{k+1}} + c_k + \frac{d_k}{p_{k+1}}}{w} = \dfrac{W_{k+1}(x) + c_{k+1}}{w} = \dfrac{U_{k+1}(x)}{w}$$
• $$H_{p_{k+1}}\left(\dfrac{x}{w}\right) \ge \dfrac{L_k(x) - \frac{U_k(x)}{p_{k+1}}}{w} \ge \dfrac{W_k(x) - \frac{W_k(x)}{p_{k+1}} - d_k - \frac{c_k}{p_{k+1}}}{w} = \dfrac{W_{k+1}(x) - d_{k+1}}{w} = \dfrac{L_{k+1}(x)}{w}$$

Does this argument hold even if existence has not been proven?

Edit 1:

Adding details based on a flag that it is unclear what I am asking. Hopefully, this helps (details as a blockquote).

Edit 2:

Shortened the question based on a comment to my question here.

If there exist $$W_k(x)$$, $$c_k$$, $$d_k$$, $$U_k(x)$$ and $$L_k(x)$$ satisfying the given recurrence relations, and
if the given inequalities hold for $$k$$, then they hold for $$k+1$$.
This does not prove that any of the inequalities hold; it does not even prove that such sequences exist. It does now suffice to prove a base case, for example by exhibiting such $$W_0(x)$$, $$c_0$$, $$d_0$$, $$U_0(x)$$ and $$L_0(x)$$.