So I've been working on this question related to standard error and I managed to show that it can't be less than $1$. But how do you show that this has to be greater than $1$ (as in cannot equal $1$)?
$$ \frac{\hat{\text{SE}}_1 + \hat{\text{SE}}_2}{\sqrt{\hat{\text{SE}}_1^2 + \hat{\text{SE}}_2^2}} $$
This is merely an algebra question. You may take $a:=\hat{\operatorname{SE}}_1>0$ and $b:=\hat{\operatorname{SE}}_2>0$. Then this expression becomes $$\frac{a+b}{\sqrt{a^2+b^2}}=\sqrt{\frac{(a+b)^2}{a^2+b^2}}=\sqrt{1+\frac{2ab}{a^2+b^2}}>1.$$