Prove that
$$x^{1116} + x^6 + \dfrac{1}{x^6+1} \ge 1$$
Below is a short proof that I have attempted. Sorry if the image causes any inconvenience as it was the clearest way to display my proof (I am not able to write proper equations (such as fractions) here yet):
I don't know if my proof is convincing enough or if it is adequate. Hence, if anybody could look at it and possibly give me any suggestions on how to improve, it would be extremely appreciated.
Thanks :)