Relative Whitney approximation for Euclidean-valued maps

Choose an open neighbourhood $U$ of $A$ on which $F$ is smooth, and then choose open sets $A\subseteq W⋐V⋐U\mathrm{.}$ Apply [L2] to each component of $F{\mathrm{\mid }}_U$ on the closed neighbourhood $\bar{V}\subseteq U$, and collect the componentwise extensions into a smooth map $G:M\to {\mathbb{R}}^k$ with $G=F$ on $\bar{V}$.

Define the continuous map $H:=F-G\mathrm{.}$ Then $H$ vanishes on $\bar{V}$. Apply [L1] to $H$ with the same error function $\epsilon \mathrm{/}2$ to obtain a smooth map $K$ satisfying $\mathrm{\parallel }K-H\mathrm{\parallel }<\epsilon \mathrm{/}2$ everywhere.

By [L3], choose a smooth cutoff $\lambda :M\to [0,1]$ with $\lambda =0$ on $W$ and $\lambda =1$ on $M\setminus V$. Set $\tilde{F}:=G+\lambda K\mathrm{.}$ On $W$ one has $\tilde{F}=G=F$. Outside $V$, one has $\tilde{F}=G+K$, so $\mathrm{\parallel }\tilde{F}-F\mathrm{\parallel }=\mathrm{\parallel }K-H\mathrm{\parallel }<\epsilon \mathrm{/}2<\epsilon \mathrm{.}$ Inside $V$, the relation $H=0$ gives $\mathrm{\parallel }\tilde{F}-F\mathrm{\parallel }=\mathrm{\parallel }\lambda K\mathrm{\parallel }\le \mathrm{\parallel }K-H\mathrm{\parallel }+\mathrm{\parallel }H\mathrm{\parallel }<\epsilon \mathrm{/}2<\epsilon \mathrm{.}$ Therefore $\tilde{F}$ is smooth, agrees with $F$ on the neighbourhood $W$ of $A$, and stays within $\epsilon$.