Convergence of n^(1/n) Sequence: Proving Convergence to 1

[nicksauce]

Homework Statement

Let $$x_n=n^{\frac{1}{n}}$$ be a sequence. Does it converge, and if so to what value?

Homework Equations

The Attempt at a Solution

Clearly it will converge to 1, so that's what we want to prove.
$$\forall \epsilon>0\exists N \textnormal{s.t.} m>N \rightarrow |m^{\frac{1}{m}}-1| < \epsilon.$$

...And then I am stuck... any hints on how to select N?

 

[jhicks]

You could try and find a weaker case for selecting N. For example, If $$\epsilon$$, N>0

$$N < (\epsilon + 1)^{N-1} < (\epsilon + 1)^{N}$$. Can you see where I'm going with this?