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**I-Think** The sequence of positive numbers $\displaystyle u_1, u_2, u_3,...$ is such that $\displaystyle u_1<4$

and

$\displaystyle u_{n+1}=\frac{5u_n+4}{u_n+2}$

By considering $\displaystyle 4-u_{n+1}$, prove by induction that $\displaystyle u_n<4$ for all $\displaystyle n\geq{1}$

My attempt N.B To make this easier to read I'm skipping all the official statements of the induction process

$\displaystyle 4-u_{n+1}=\frac{4-u_n}{2+u_n}$

Let $\displaystyle n=1$

$\displaystyle 4-u_2=\frac{4-u_1}{2+u_1}$

It is given that $\displaystyle u_1<4$ and positive so $\displaystyle \frac{4-u_1}{2+u_1}<4$

$\displaystyle u_2=4-\frac{4-u_1}{2+u_1}$

Hence $\displaystyle u_2$ is less than $\displaystyle 4$

Assume true for $\displaystyle n=k$

$\displaystyle 4-u_{k+1}=\frac{4-u_k}{2+u_k}$

Test at $\displaystyle n=k+1$

End of attempt

To finish this step confuses me. I implore the forum for assistance.