inequality about $n^{1/n}$ When I was studying application of the sandwich theory, it was concluded that $n^{1/n}>1$ with regard to prove $\lim_{n\to\infty}n^{1/n}=1$.
In my opinion, it is obviously warranted that $n^{1/n}>0$. I think $n^{1/n}>1$ is certain only when $n>1$, but we can't be of the certain value of $n$. Does the $\lim_{n\to\infty}$ determine that $n^{1/n}>1$?
 A: If the notation $\lim_{n\to \infty}$ is used, it is understood that $n$ is a natural number. For $n=0$, the expression is undefined, and for $n=1$, it is trivially false, so yes, it is implicitly assumed that $n>1$.
Note that in the context of limits, $a_n>1$ is often only interesting for sufficiently large $n$.
A: When we write mathematics, we usually stick to certain conventions. One is that $n$ representas a natural number, i.e. $n\in \Bbb N$. Other variants can be $k$,$i$, $j$, but those are mostly used in sums, for the indices. So when one writes
$$\lim_{n \to \infty} n^{\frac 1 n }$$
one is implicitly asking for the limit of the sequence $a_n = n^{\frac 1 n }$. It'd be very odd for someone to write $n$ if they want $n$ a real number. If $n$ is real, we would usually use $y$, $x$, $r$, $q$, or specify that indeed it is real. Even so, the use of $n$ would be avoided, since most of us have our brains already programmed after some time of math reading.
Just for the sake of it, you will often see
$$j,k,l,m,n \in \Bbb N$$ 
$$p,q,r \in \Bbb Q$$ 
$$a,b,c,d,x,y,z \in \Bbb R$$
$$x,y,z \in \Bbb C$$
I'm not saying this is strict (as for example Apostol uses $x$ and $y$ for integers, but he explicitly states that they are integers.) but it will help you when you write, for others to interpret what you're writing, and for you to read maths with more ease. 
