N less than M to the N

Theorem

$\forall m, n \in \Z_{>0}: m > 1 \implies n < m^n$

Proof

\(\ds n\)

\(=\)

\(\ds \underbrace {1 + 1 + \cdots + 1}_{\text {$n$ times} }\)

\(\ds \)

\(<\)

\(\ds 1 + m + m^2 + \cdots + m^{n - 1}\)

as $m > 1$

\(\ds \)

\(=\)

\(\ds \frac {m^n - 1} {m - 1}\)

Sum of Geometric Sequence

\(\ds \)

\(\le\)

\(\ds m^n - 1\)

as $m - 1 \ge 1$

\(\ds \)

\(<\)

\(\ds m^n\)

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {1-1}$ Principle of Mathematical Induction: Corollary $\text {1-1}$