Please provide assistance for the proof that the error between the function $W(\lambda(\sqrt{n}))$ and the fraction of composite integers in $\mathbb{O}_n$ declines as $n$ increases.
Let $\mathbb{O}_n$ represent the set of odd integers less than or equal to $n$ excluding 1.
$\mathbb{O}_n$ = {3,5,7,9,11,13,15,17,19,…n}
Let the function $l(x)$ equal the largest prime number less than $x$.
For example: $l(10) = 7, l(20) = 19$ and $l(23) = 19$
Let the function $\lambda(x)$ equal the largest prime number less than or equal to $x$.
For example: $\lambda(10) = 7, \lambda(20) = 19$ and $\lambda(23) = 23$
The function $W(\lambda(\sqrt{n}))$ was derived to represent the fraction of composite integers in the set $\mathbb{O}_n$ for large values of $n$.
\begin{eqnarray*} W(\lambda(\sqrt{n})) = \sum_{\substack{p=3\\p\ \text{prime}}}^{\lambda(\sqrt{n})}\left((\frac{1}{p})\prod_{\substack{q=3\\q\ \text{prime}}}^{l(p)}\frac{(q-1)}{q}\right) \end{eqnarray*} or
$W(\lambda(\sqrt{n}))= \frac{1}{3} + (\frac{1}{5})(\frac{2}{3}) + (\frac{1}{7})(\frac{2}{3})(\frac{4}{5}) + ... + (\frac{1}{ \lambda(\sqrt{n})})(\frac{2}{3})(\frac{4}{5})(\frac{6}{7})(\frac{10}{11})...(\frac{(l(\lambda(\sqrt{n})) - 1)}{l(\lambda(\sqrt{n}))})$.\
I have proven that the maximum error between $W(\lambda(\sqrt{n}))$ and the actual fraction of composite integers in the set $\mathbb{O}_n$ declines as $n$ increases in sections 6 and 7 in this paper http://vixra.org/pdf/1901.0436v5.pdf.
Is this proof sufficient? If not, what needs to be done?
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