Discussion on answer by Pixel Pirate: Proof by contradiction: every real number $x \in[0,\frac { \pi} 2]$ we have $\sin x+\cos x \geqslant 1$

Imported from a comment discussion on https://math.stackexchange.com/questions/4979680/proof-by-contradiction-every-real-number-x-in0-frac-pi-2-we-have-si/4979683#4979683