\[\boxed{\text{252}\text{\ (252)}\text{.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[\textbf{а)}\ x^{n + 1} - x^{n} = x^{n} \cdot (x - 1);\]
\[так\ как\ x \in \lbrack 0;1\rbrack,\ \]
\[то\ x - 1 \leq 0 \Longrightarrow\]
\[\Longrightarrow x^{n} \cdot (x - 1) \leq 0 \Longrightarrow\]
\[\Longrightarrow x^{n + 1} \leq x^{n}.\]
\[\textbf{б)}\ x^{n + 1} - x^{n} = x^{n} \cdot (x - 1);\]
\[так\ как\ x \in (1;\ + \infty),\ \]
\[то\ x - 1 > 0 \Longrightarrow\]
\[\Longrightarrow x^{n} \cdot (x - 1) > 0 \Longrightarrow\]
\[\Longrightarrow x^{n + 1} > x^{n}.\]