Your induction hypothesis is that $3\le a_k<4$, and you want to use that to show that $3\le a_{k+1}<4$. This is just a matter of checking some simple algebra of inequalities: you know that $a_{k+1}=4-\frac2{a_k}\;,$ and you’re assuming that $3\le a_k<4$. How can you use that assumption to get information about $a_{k+1}$?
Start by seeing what it says about $\dfrac2{a_k}$: if $3\le a_k<4$, then
$\frac13\ge\frac1{a_k}>\frac14\;,$ so $\frac23\ge\frac2{a_k}>\frac24=\frac12\;.$
Now what does that say about $a_{k+1}=4-\dfrac2{a_k}$? Once again the inequalities have to be inverted, and we get
$4-\frac23\le4-\frac2{a_k}<4-\frac12\;.$
I’ll leave the last little bit of cleaning up to you.