The question gives a Utility function for a bundle contains two goods, $$U(x_{1}, x_{2}) = \sqrt{x_{1}} + \sqrt{x_{2}}$$
I am trying to prove that the preference represented by this utility function is convex. This is what I got so far:
Let $\lambda \in[0,1], x, y \in \succsim(x), y \succsim x$. If $U(\lambda x+(1-\lambda) y) \geqslant U(x)$, then the preference is convex.
So $\sqrt{\lambda x_{1}+(1-\lambda) y_{1}}+\sqrt{\lambda x_{2}+(1-\lambda) y_{2}} \geqslant \sqrt{x_{1}}+\sqrt{x_{2}}$ needs to be true.
Since $\lambda \in[0,1]$,
$$\sqrt{\lambda x_{1}+(1-\lambda) y_{1}}+\sqrt{\lambda x_{2}+(1-\lambda) y_{2}} \geqslant \lambda\left(\sqrt{x_{1}}+\sqrt{x_{2}}\right)+(1-\lambda)\left(\sqrt{y_{1}}+\sqrt{y_{2}}\right)$$
But then I am confused about how to show that the inequality is true. I am thinking about comparing component-wise, and find difficult to show that
$$\sqrt{\lambda x_{1}+(1-\lambda) y_{1}} \geqslant \lambda \sqrt{x_{1}} + (1-\lambda)\sqrt{y_{1}}$$
and $$\sqrt{\lambda x_{2}+(1-\lambda) y_{2}} \geqslant \lambda \sqrt{x_{2}} + (1-\lambda)\sqrt{y_{2}}$$
I would really appreciate if anyone can give me some hints on this, thank you so much!