I have a question about the Schwartz space integrity test in Folland, proposition 8.2.

Drink $ (f_k) $ be a Cauchy sequence in Schwartz's space $ S $.

I understand that in the test you built $ g_0 $ that satisfies

$$ partial ^ alpha f_k to partial ^ alpha g_0 $$

evenly But in the definition of the norm $ | dot | _ (N, α) $there is a factor $ (1+ | x |) ^ N $ and after taking the sup, how can we guarantee the uniform convergence of

$$ (1+ | x |) ^ N ( partial ^ alpha f_k) a (1+ | x |) ^ N ( partial ^ alpha g_0)? $$