Show a sequence (Un) converges to $0$ in Normed space $(V, || ||)$. Where $U_{n+1} = U_{n}*sin(||Un||)$. And ||U_0|| = a. a is an element of [0,pi/2)

Question

I managed to prove that ||U_n|| is a decreasing sequence. By showing |sin(||Un||)|<1.

I am struggling to prove that the sequence converges to the 0 vector in the Normed Space. Any ideas? I found it would be sufficent to show ||Un|| converges to 0 in the Reals.

Answer 1

Here's a hint:

The sequence $(\|u_n\|)$ is decreasing and bounded below, hence convergent to some value $\alpha$. Using continuity, show that $$\alpha=\alpha\sin\alpha.$$ And assuming $\alpha\neq0$, you obtain a contradiction, so we must have $\alpha=0$.