Multivariable Chain Rule: Proof

Note: There's a slight error at 25:00.
I wrote |Dg(a)||h| + |E(h)| is less than |Dg(a)|h. It should instead be |Dg(a)||h| + |E(h)| is less than |Dg(a)||h| + L|h|, where L is some arbitrary constant. This holds whenever |h| is sufficiently small, as we know |E(h)| approaches 0 as |h| approaches 0.