The statement of Local Immersion Theorem as follows:
Suppose that $f:M\rightarrow N$ is an immersion at $x\in M$ and $y\in N$. Then there exists local coordinates around $x$ and $y$ such that $$f(x_1,\dots ,x_k)=(x_1,\dots,x_k,0,\dots,0)$$ In other words f is locally equivalent to the canonical immersion.
I understood the proof completely. But I have a problem in the statement itself. I could prove that f is locally equivalent to canonical immersion.
But how does that imply the existence of the coordinates around x and y such that f takes the above form.
Edit: I could prove that there exist $\phi: U\rightarrow U'$ and $\psi: V\rightarrow V'$ where $x\in U',y\in V'$ such that $\psi^{-1}f\phi=cannonical~immersion$.
But how does that imply there exist local co ordinates such that $$f(x_1,\dots ,x_k)=(x_1,\dots,x_k,0,\dots,0)$$.
Please let me know if you still do not get my doubt.
The point is that the parametrizations $\phi$ and $\psi$ give local coordinate systems $\phi^{-1}$ (on $U'$) and $\psi^{-1}$ (on $V'$) in which that equation holds. $(x_1,\dots,x_k)$ are the coordinates of $\phi^{-1}$ and $(y_1,\dots,y_\ell)$ are the coordinates of $\psi^{-1}$. This language appears throughout differential topology/geometry, and so you need to get used to it!