I am studying for an exam and going through various earlier tutorial sheet questions. For the question below, I have tried and just can't figure out how to prove that $x$n$ $ < $ 3$ by mathematical induction. Does anyone know how to prove this?
The question is:
Let $x$1 = 1, and for each $n \in \mathbb{N}$ let $x$n+1 = $\frac{2}{3}x$n$ $ + $ 1$. Then $x$n$ $ < $ 3$ for all $n \in \mathbb{N}$.
Hint : For the inductive step, you have
$$\begin{align}x_{n + 1} &= \frac{2}{3}x_n + 1\\&<\frac{2}{3}(3) + 1\\&= 3\end{align}$$