I came across two different definition of differentiability of a function of several variable at a point.
1) (due to Spivak) $f:\mathbb R^n\to\mathbb R^m$ is differentiable at $a$ if for some linear function $\lambda:\mathbb R^n\to\mathbb R^m$ we have $\lim_{h\to0}\frac{|f(a+h)-f(a)-\lambda(h)|}{|h|}=0$
2) (found here) $f:\mathbb R^n\to\mathbb R^m$ is differentiable at $a$ if for some linear function $\lambda:\mathbb R^n\to\mathbb R^m$ we have $\lim_{h\to0}\frac{f(a+h)-f(a)-\lambda(h)}{|h|}=0\in\mathbb R^n$
Are these two definitions equivalent? I can see 2 implies 1, but what about the reverse inclusion?
In case they are not equivalent which one should be taken as correct?