Jordan measurable

I am trying to show that a rectifiable curve in R^n has jordan content
zero.

I can form a partition on the curve, and then cover the curve by a
finite number of n-boxes. Intuitively, as I refine the partition, the
volume of the boxes gets smaller and smaller and should approach some
limit (zero) since the curve is rectifiable. However, how do I prove
this rigorously?

.

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