Ta có:

• \[\frac{x}{{x + y + z}} > \frac{x}{{x + y + z + t}}\]

• \[\frac{y}{{x + y + t}} > \frac{y}{{x + y + z + t}}\]

• \[\frac{z}{{y + z + t}} > \frac{z}{{x + y + z + t}}\]

Do đó \[M > \frac{x}{{x + y + z + t}} + \frac{y}{{x + y + z + t}} + \frac{z}{{x + y + z + t}} + \frac{t}{{x + y + z + t}}\]

\[ \Rightarrow M > \frac{{x + y + z + t}}{{x + y + z + t}}\]

Þ M > 1

• \[\frac{x}{{x + y + z}} < 1 \Rightarrow \frac{{x + t}}{{x + y + z + t}} > \frac{x}{{x + y + z}}\]

• \[\frac{y}{{x + y + t}} < 1 \Rightarrow \frac{{y + z}}{{x + y + z + t}} > \frac{y}{{x + y + t}}\]

• \[\frac{z}{{y + z + t}} < 1 \Rightarrow \frac{{x + z}}{{x + y + z + t}} > \frac{z}{{y + z + t}}\]

• \[\frac{t}{{x + z + t}} < 1 \Rightarrow \frac{{y + t}}{{x + y + z + t}} > \frac{t}{{x + z + t}}\]

\[ \Rightarrow M < \frac{{2\left( {x + y + z + t} \right)}}{{x + y + z + t}}\]

Þ M < 2

Ta có: 1 < M < 2 Þ M không phải là số tự nhiên