Spaces of smooth sections of tensor bundle are infinite dimensional vector spaces I have been reading tensor fields from Introduction to Smooth Manifolds by John Lee.
A bundle of covariant k-tensors on a smooth manifold M  given by $ T^{k}T^{*}M $ is called a tensor bundle over M
A section of this bundle is a covariant tensor field on M
The space of smooth sections of $ T^{k}T^{*}M $ denoted by $ \Gamma (T^{k}T^{*}M) $ is an infinite dimensional vector space over $\mathbb{R}$
My question is that i am unable to see that $ \Gamma (T^{k}T^{*}M) $ is an infinite dimensional vector space. 
I can see that addition of two tensor fields is a tensor field (hence a smooth section) and multiplication of a tensor field by a real number, gives a tensor field. So $ \Gamma (T^{k}T^{*}M) $ is a vector space. Not sure how to see that its infinite dimensional?

Take an open $U \subset M$ over which the bundle is trivial, and let $\beta$ be a nowhere vanishing section over $U$. You can embed the space $\mathscr{D}(U)$ of smooth functions with compact support in $U$ into the space of global sections via the map
$$\varphi \mapsto \varphi \cdot \beta \colon x \mapsto \varphi(x)\cdot \beta(x)\,.$$
Since $\mathscr{D}(U)$ is infinite-dimensional for $U \neq \varnothing$ (if the dimension is greater than $0$), it follows that the space of global sections of the bundle is infinite-dimensional for a manifold $M$ of positive dimension.
