finding the returns to scale of a production function
What you have proven is that $$f(tx_1, tx_2) &gt; tg(x_1,x_2)$$

which translates "scaled $f$ is higher than some other, homogeneous function $g$ scaled by the same factor". This does not prove anything about the returns to scale related to $f$, although I can see why it may appear otherwise.

You can go in reverse, starting from 

$$tf(x_1,x_2) = t^{0.5}t^{0.5}f(x_1,x_2)=...$$

and compare to $f(tx_1, tx_2)$. 

Also beware about the following: what are the permissible values for $t$ when returns are not constant? Are you sure we should obtain the result for $t&gt;0$, or maybe we examine the property for a smaller interval?