<s>[INST] prove $ 1 < \varphi = \frac{1 + \sqrt{5}}{2} < 2$ Prove $ 1 < \varphi =  \frac{1 + \sqrt{5}}{2} < 2$
Just want to see if my "reasoning is sound.
1) Showing $1 < \frac{1 + \sqrt{5}}{2}$
Consider $\frac{1}{2}$:
$$\frac{1}{2} < \frac{1}{2} + \frac{1}{2} < \frac{1}{2} + \frac{\sqrt{5}}{2} = \frac{1 + \sqrt{5}}{2} $$
2) Showing $\frac{1 + \sqrt{5}}{2} < 2$
We know $$\sqrt{5} < 3 \\ \Rightarrow 1 + \sqrt{5} < 4 \\ \Rightarrow \frac{1 + \sqrt{5}}{2} < 2$$
Therefore: $ 1 < \varphi =  \frac{1 + \sqrt{5}}{2} < 2$
 [/INST] Another way
As $\varphi>0,$
$\varphi=1+\dfrac1{\varphi}>1$ 
$\implies1+\dfrac1{\varphi}<1+1$
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