  We show that the Teichm\"uller space of a surface without boundary and with
punctures, equipped with Thurston's metric is the limit (in an appropriate
sense) of Teichm\"uller spaces of surfaces with boundary, equipped with their
arc metrics, when the boundary lengths tend to zero. We use this to obtain a
result on the translation distances for mapping classes for their actions on
Teichm\"uller spaces equipped with their arc metrics.
