1. ## Induction on sequences

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.

2. Originally Posted by 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.
You're so close!

You have a nice way of writing 4-u_{k+1} in terms of u_k. So now you need to show that $\displaystyle \frac{4-u_k}{2+u_k}<4$, which in turn shows that 4-u_{k+1}<4. This should be easy for you, since you have assumed that u_k < 4. Use that to show that the above fraction is < 4, thus 4-u_{k+1}<4, thus u_{k+1}<4. This shows that u_{k} < 4 implies u_{k+1}<4, which is the proof.