Number to Reciprocal Power is Decreasing

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

Proof
We want to show that $\left({n + 1}\right)^{1/\left({n + 1}\right)} \le n^{1/n}$.

Thus:

But from One Plus Reciprocal to the Nth, $\left({1 + \dfrac 1 n}\right)^n < 3$.

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

So $\left \langle {n^{1/n}} \right \rangle$ is decreasing for $n \ge 3$.