Limit of Integer to Reciprocal Power/Proof 1

Theorem

Let $\sequence {x_n}$ be the real sequence defined as $x_n = n^{1/n}$, using exponentiation.

Then $\sequence {x_n}$ converges with a limit of $1$.

From Number to Reciprocal Power is Decreasing we have that the real sequence $\sequence {n^{1/n} }$ is decreasing for $n \ge 3$.

Now, as $n^{1 / n} > 0$ for all positive $n$, it follows that $\sequence {n^{1 / n} }$ is bounded below (by $0$, for a start).

Thus the subsequence of $\sequence {n^{1 / n} }$ consisting of all the terms of $\sequence {n^{1 / n} }$ where $n \ge 3$ is convergent by the Monotone Convergence Theorem (Real Analysis).

Now we need to demonstrate that this limit is in fact $1$.

Let $n^{1 / n} \to l$ as $n \to \infty$.

Having established this, we can investigate the subsequence $\sequence {\paren {2 n}^{1 / {2 n} } }$.

From Limit of Root of Positive Real Number, we have that $2^{1 / {2 n} } \to 1$ as $n \to \infty$.

So $n^{1 / {2 n} } \to l$ as $n \to \infty$ by the Combination Theorem for Sequences.

Thus:

$n^{1 / n} = n^{1 / {2 n} } \cdot n^{1 / {2 n} } \to l \cdot l = l^2$

as $n \to \infty$.

So $l^2 = l$, and as $l \ge 1$ the result follows.

$\blacksquare$