Limits- legal and illegal algebraic manipulation. I am doing probability and am using the strong law of large numbers to get after a bit of irrelevant extra algebra which I wont mention 
$\frac{\log(C_n)}{n} \rightarrow K$ where $C_i$ are a sequence of random variables, $K$ is some constant and $n$ can only take natural number values.
I want to do the following:
$\Rightarrow$ $\log(C_n) \rightarrow nK$
$\Rightarrow$ $(C_n) \rightarrow e^{nK}$
This is definitely not allowed I know- but I'm trying to figure out how $C_n$ behaves and I don't know what to do??
 A: While you can't say that $\log(C_n)\to nK$ and $C_n \to e^{nK}$, your intuition is right -- you can make it precise. 
The idea behind your assertions makes sense. If $\log(C_n)/n$ approaches $K$, then for large $n$, we should have $\log(C_n)$ approximately equal to $nK$. Thus we should have $\log(C_n)$ approximately equal to $e^{nK}$.
More precisely, what does this mean? It means that there is a function $\epsilon(n)$ such that $\epsilon(n)/n \to 0$ and
$$ K - \epsilon(n)/n \leq \log(C_n)/n \leq K + \epsilon(n)/n. $$
Now we can do algebra all day long. 
$$ Kn - \epsilon(n) \leq \log(C_n) \leq nK + \epsilon(n) $$
and exponentiating (which is monotone)
$$ e^{Kn}e^{-\epsilon(n)} \leq C_n \leq e^{Kn}e^{\epsilon(n)} $$
This actually gives more information than you asked for. It tells you not just that $C_n$ behaves like $e^{nK}$ for large $n$, it tells you how quickly $C_n$ tends to behave like that.
A: If
$\frac{\log(C_n)}{n} \rightarrow K
$,
then
$|\frac{\log(C_n)}{n} - K|
< \epsilon
$
for all large enough $n$.
Multiplying by $n$,
$|\log(C_n) - nK|
< n\epsilon
$,
or
$-n\epsilon
<\log(C_n) - nK
< n\epsilon
$.
Exponentiating,
$e^{-n\epsilon}
<C_ne^{- nK}
< e^{n\epsilon}
$,
or
$e^{nK-n\epsilon}
<C_n
< e^{nK+n\epsilon}
$,
or
$e^{n(K-\epsilon)}
<C_n
< e^{n(K+\epsilon)}
$,
and that is about 
all you can do.
It would be nice
is you could first choose $n$
and then make
$\epsilon$ small,
but it does not
work that way.
You first choose
$\epsilon$
and
then you find the
values of $n$
for which the inequality holds.
