This problem comes is from Hoffman's Analysis in Euclidean Space.
True or False. Let $\{X_n\}$ be a sequence in $\mathbb{R}^m$. If $\{X_{n}\}$ converges to a (nonzero) vector $X$, does the angle between the vectors $X_n$ and $X$ converge to $0$?
I am assuming it is true. Here is my attempt at a proof.
Proof. Assume that $\{X_{n}\}$ converges to the vector $X$. We know that $$\cos\theta=\frac{\langle X,X_n\rangle}{|X||X_n|}$$ for some positive integer $n$. Since $$\lim\limits_{n\rightarrow\infty}\frac{\langle X,X_n\rangle}{|X||X_n|} =\frac{|X|^2}{|X|^2}=1$$ then $$\theta=\cos^{-1}(1)=0$$
I can't see a reason why the converse statement wouldn't be necessarily true. In other words, if the angle between ${X_{n}}$ and some vector $X$ approaches zero, wouldn't the sequence $\{X_n\}$ converge to $X$?
EDIT: Another thing that popped into my mind. Do these results hold in general for all vector spaces?
Your proof is correct.
The converse is not true because the angles do not speak to the norms. The angles may converge to a limit but the vectors do not have to converge due an oscillation in norms.