<s>[INST] Proving that (4-2/1)(4-2/2)...(4-2/n) in an integer. We have to prove that $(4-2/1)(4-2/2)...(4-2/n)$ is an integer for $n\in\mathbb{N}$. Can we do this by induction?
We prove for $n = 1$, which is trivial as $(4-2/1) = 2$ which is clearly an integer.
next we assume that the statement is true for some integer $n$
i.e. that $(4-2/1)(4-2/2)...(4-2/n) = p$ where p is an integer, and we prove for $n+1$.
So we need to show that $(4-2/1)(4-2/2)...(4-2/n)(4-2/(n+1))$ is also an integer. Which is the same as saying $p\times(4-\frac{2}{n+1})$ is an integer. 
We can rewrite $p$ as $\dfrac{2^n(1\times3\times5\times...\times(2n-1))}{n!}$ so essentially all we need to prove is that $n+1$ divides the numerator of $p$. So assume that $n$ is even, which implies that $n+1$ is odd and certainly less than $2n-1$ so therefore it will divide one of the odd numbers in the numerator of $p$. If $n$ is odd, then $n+1$ is even and will not have more factors of $2$ than $2^n$. So if the prime factorization of $n+1$ contains $k$ factors of 2, $k<n$ or $k=n$,  implying that $2^k$ divides $2^n$. Also the prime factorisation of $n+1$ will not have have an odd factor that is greater than $2n-1$ as it already has a factor of 2. Therefore $n+1$ divides the numerator of p. 
Can we do this? It seems like it is not entirely correct! I may have made a big mistake in my reasoning. 
Any help is appreciated.
 [/INST] [Cannot seem to add a comment (re request above on the induction method), so posting a new answer in response]
Assume result is true for $n$ terms, i.e. product is 
$$2n\choose n$$ 
which is an integer.
Then, for $n+1$ terms, product becomes
$$\begin{align}
{2n\choose  n}\left( 4-\frac 2{n+1}\right) &=\frac{4n+2}{n+1} {2n\choose n} \\
&=2\cdot \frac{2n+1}{n+1}\cdot {2n\choose n} \\
&=\frac {2n+2}{n+1}\cdot \frac{2n+1}{n+1}\cdot\frac{2n(2n-1)...(n+2)(n+1)}{1\cdot2\cdot ... n} \\
&=\frac {(2n+2)(2n+1)(2n)...(n+2)}{1\cdot 2\cdot ... n(n+1)}\\
&={{2n+2}\choose{n+1}}\\
&={{2(n+1)}\choose{n+1}}\\
\end{align}$$
i.e. also true for $n+1$.
It can easily be shown that the product is true for $n=1$. 
Hence, by induction, the proposition is true for all integer $n$.
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