How to prove the statement:
Assuming the continuum hypothesis, the product of any uncountable family of $T_1$ spaces, each having more than one point, is never sequentially compact.
The statement appears in General Topology by Stephen Willard, exercise 17G.6.
My attempt was along the lines of trying to use the sequential compactness (any sequence of points in the space has a convergent subsequence whose limit is in it) of the space defined to build a set that would violate the continuum hypothesis. But I could not find the right construct.