Weak approximation for inequivalent absolute values ($K\text{ is dense in }{\prod }_{i=1}^r{K}_{v_i}$)

For finitely many pairwise inequivalent nontrivial absolute values on a field, an element of the field can approximate independently prescribed targets in the associated completion of a valued field at each place to arbitrary positive accuracies. Equivalently the diagonal map has dense image in the finite product. Requiring finitely many inequivalent places is essential; the conclusion controls finitely many inequalities, not infinitely many independently prescribed bounds at once, or incompatible targets at one place. Weak approximation for number fields is the class-field-theoretic instance.

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, $|y_n|\le 1/n!\to 0$. 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 $|y_n{|}_p\to 0$.

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.