Lemma 2.4.2 Convergence in product spaces Let (X, d) and (Y, e) be met- Hc spaces, and let (zn) be a sequence in the product space Xx Y. (Thus each term of zn of the sequence is of the form zn. and yn EY.) Then the following are equivalent: (xn, Yn), where xn E X 1. The sequence (zn) converges to z = (x,y) E X x Y. 2 The sequence (xn) converges to x E X and the sequence (yn) converges to y E Y.

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