I am trying to show that if there exists a global continuous frame for a smooth vector bundle over a smooth manifold $M$, then we can find a global smooth frame.
First, I have proven the simple situation when the vector bundle is $M \times \mathbb{R}^k$.
Next, I want to use partition of unity to prove it for the general case. But I am stuck here because when I sum up the local smooth frames, it is not true that the sum is still a frame. So I want to use some tricks to make it still linearly independent for the sum at every points but I have no idea. Can somebody help me?