= Rational sequence converging to zero at every standard absolute value
{title2=$y_n=\dfrac{n!}{1+(n!)^2}$}
The displayed nonzero rational sequence tends to zero both in the usual real <absolute value> and in every <p-adic absolute value>. In the real absolute value, $|y_n|\le1/n!\to0$. For a fixed prime $p$ and $n\ge p$, the denominator is congruent to one modulo $p$, so its <P-adic valuation> is zero. Therefore $v_p(y_n)=v_p(n!)\to\infty$, proving $|y_n|_p\to0$.
This is convergence at each fixed place; it is not uniform convergence over all primes. For each particular $n$, new prime divisors of the denominator can compensate the small values at the previously fixed places, consistently with the <product formula>. Thus a product identity for a single rational number does not prohibit pointwise convergence at infinitely many absolute values.
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