Sobolev Embeddings Part 2 - Nonlinear Analysis on Manifold

Question

I was studying the proof of Theorem 3.4 from the book Nonlinear analysis on Manifolds - Sobolev Spaces and Inequalities by Emmanuel Hebey. I don't know how to conclude this part. The author defines a bounded, compacted supported Lipschitz function $\alpha_{i}$ and then, sets: $$\eta_{i} = \frac{\alpha_{i}^{[q]+1}}{\sum_{m} \alpha_{m}^{[q]+1}}$$ where $q>n$ (the dimension of the manifold), $[q]$ is the integer part of $q$, supp $\alpha_{i} \subset B_{x_{i}}(r_{H})$, where $(B_{x_{i}}(r_{H}))_{i}$ is an open, uniformly, locally finite covering of $M$. The proof follows by saying that $\eta_{i}$ and $\eta_{i}^{1/q}$ are both Lipschitz. I was trying to derive that from the definition but I was not able to conclude. Could anyone help me to clarify that? Even a small tip would be very helpful. I appreciate any help you can provide.