I am trying to understand the following theorem from Lee's book about Riemannian geometry. I had a look at several books but unfortunaletly I am still not quite sure if I get it.
Question 1: Why does he assume w.l.o.g. that $X_{p}=0$. I would say that the proof applied to $X(p)-\tilde{X}(p)=0$ gives the disered result, but why can't it be done "directly" as in the following excerpt. I would have done it this way too.
Question 2: I had a look at Boothby's book and what I dont't understand is why he constructs a vector field that agrees with $X$ in a nhd. Is it just supposed to be a proof of existence in the sense that we can actually find such a vector field? Furthermore could it be that there are some typos? For example in the the equation above the last paragraph.
I know it is quite a long question bit I would really appreciate some help. Many thanks in advance.