Approximation of elements in henselizations

For valued fields $K$ of rank higher than 1, we describe how elements in the henselization $K^h$ of $K$ can be approximated from within $K$; our result is a handy generalization of the well-known fact that in rank 1, all of these elements lie in the completion of $K$. We apply the result to show that if an element $z$ algebraic over $K$ can be approximated from within $K$ in the same way as an element in $K^h$, then $K(z)$ is not linearly disjoint from $K^h$ over $K$.

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