<s>[INST] Let $a^2<2, b=2(a+1)/(a+2)$. Show $b^2<2$ (assignment) It is a part of my assignment. 
$$ \text {Let }a^2<2, \quad b=2\frac {(a+1)}{(a+2)}\quad \text{ Show } b^2<2$$ 
I already proved that a

But, I am struggling to prove $b^2<2$. 
My lecturer said that I need to manipulate $b^2$, which is larger than $b^2$ but less than 2. That is $4(a+1)^2/(a+2)^2$ < some number < 2.  
I was trying this for hours, but I couldn't find the way to solve. Thanks for helping in advance. 
 [/INST] Hint: $\require{cancel}\;b^2 - 2 = \dfrac{4(a+1)^2}{(a+2)^2}-2=\dfrac{4a^2+\cancel{8a}+4-2a^2-\cancel{8a}-8}{(a+2)^2}=\dfrac{2a^2-4}{(a+2)^2}=\dfrac{2(a^2-2)}{(a+2)^2}\,$
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