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Let $f(x)$ be a function defined on $\mathbb{R}$ as $$f(x)=\sin \frac{\pi x}{2}.$$ For $y\in \mathbb{R}$, consider the sequence defined by $$x_0(y)=y\ \ \text{and}\ \ x_{n+1}(y)=f(x_n(y)) \ \text{for all}\ n\geq 1.$$ Let$$g(y)=\lim_{n\to\infty} x_n(y)\ \text{for}\ y\in [0, 3].$$Then show that $$\...
