Induction proof inequality

So I got this induction proof question but I can't seem to make a logical statement in one part of it:

The question is , $a_{n + 1} = 5 - \frac{6}{a_n + 2}$ with
$a_1 = 1$ . Prove by induction that $a_n < 4$ for $n \geq 1$

I reached up to the proof where I need to prove $a_{k+1} <4$

Proof

$a_k <4 \implies a_k + 2<6$

The next step I want to put is:

$\frac{6}{ a_k +2} >1$

However I can only justify this statement if $a_k > -2$ but I can't seem to prove that or find any info in the question to suggest that it.

Can anyone help me with the proof or my theory?