Number to Reciprocal Power is Decreasing

Theorem
The real sequence $\sequence {n^{1/n} }$ is decreasing for $n \ge 3$.

Proof
We want to show that $\paren {n + 1}^{1 / \paren {n + 1} } \le n^{1/n}$.

Thus:

But from One Plus Reciprocal to the Nth:
 * $\paren {1 + \dfrac 1 n}^n < 3$

Thus the reversible chain of implication can be invoked and we see that $\paren {n + 1}^{1 / \paren {n + 1} } \le n^{1/n}$ when $n \ge 3$.

So $\sequence {n^{1 / n} }$ is decreasing for $n \ge 3$.