I everyone, here there is the proof of the Peano Theorem, but I don't understand when the book says
Now Let $K$ be so large that for all $k>K$, $$\mid \varphi (x) - \varphi^{(k)}(x)\mid <M\eta$$ and all links of the polygonal arc $L_{k}$ have length less than $\eta$. Then if $\mid x' - x\mid < 2\eta$, all the Euler polygons $y=\varphi^{(k)}(x)$ for which $k>K$ lie entirely in the interior of $Q$.
The first inquality makes sense since $$\lim_{k\rightarrow +\infty}\varphi^{(k)}(x)=\varphi (x),$$ and also the fact that every link has length less than $\eta$ by the construction of $L_{k}$. My problem is that I can't see why with this hypothesis $\mid y'-y\mid < 4M\eta$. Can someone help me? Thanks before.
By construction, the Euler polygons consist of straight line segments with gradient $<M$. This limits the y-variation of the polygon for a given x-variation. We also have a condition limiting how far from $y'$ the polygon passes when $x=x'$.
Putting these together, $$\mid y'-y\mid = \mid \varphi(x')-\varphi^{(k)}(x)\mid$$ $$\le\; \mid \varphi(x')-\varphi^{(k)}(x')\mid + \mid \varphi^{(k)}(x')-\varphi^{(k)}(x)\mid$$ $$\lt M\eta + M\mid x'-x\mid$$ $$= M\eta +2M\eta$$ $$\lt 4M\eta$$
It would probably help to draw a picture at this point!