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I know that ##|x-x_0|<\epsilon/5##, and that ##|y-y_0|<\epsilon\Leftrightarrow|y-y_0|/5<\epsilon/5##, but I don't think that that suffices to conclude ##|x-x_0|\leq|y-y_0|/5##.

Any thoughts? Thanks.... indeed ##|x-x_0|## is sufficiently small if it does not exceed one-fifth of the value of ##|y-y_0|##.