real analysis – Can we define a metric on the Hilbert cube using any lp metric?

This is the definition given for the Hilbert Cube:
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The metric defined here is basically just the L2 metric. I was wondering if we can use any of the Lp metric instead as long as p>1. It can be shown using the comparison test that they all converge, so any of those metric should also be a well-defined metric correct?