Approximation to any real number by rationals Given a real number $x$, one can show that there exists infinitely many $p,q$, such that $\gcd(p,q)=1$, and $|x-\frac{p}{q}|\leq \frac{1}{q^2}$. (There is a hint saying that I should use the pigeon hole principle, but how?)
And that for any $\varepsilon>0$, there are only finitely many $p,q$, such that $\gcd(p,q)=1$, and $|x-\frac{p}{q}|\leq \frac{1}{q^{2+\varepsilon}}$
I am very confused by this problem, can anyone help me with it? Thanks.
 A: You are asking about one of the the basic problems of Diophantine Approximation, which is an important subfield of number theory.  (Thus I disagree with Christian Blatter's answer that these are isolated results, although I suppose this depends a lot on how you want to group results together and gerrymander various subfields of mathematics.)  Other results in this area include the Siegel-Mahler S-Unit Theorem and Siegel's Big Theorem on S-integral points on algebraic curves.
Both of your claims are false as stated: there are missing hypotheses.
Your first claim fails for rational numbers: if $x$ is rational, there is one excellent rational approximation -- namely, take $\frac{p}{q} = x$ -- but one cannot find infinitely many distinct rational numbers $\frac{p}{q}$ such that $|x-\frac{p}{q}| \leq \frac{1}{q^2}$.  In fact, we cannot do this with $\frac{1}{q^2}$ replaced by $\frac{1}{q^{1+\epsilon}}$ for any positive $\epsilon$.
However the result is true if $x$ is irrational.  The proof is extremely elementary and uses no more than the Pigeonhole Principle.  In fact this is the result for which the Pigeonhole Principle was first explicitly identified as a useful proof technique.  Note that in certain circles PHP is also called "Dirichlet's Box Principle" or "Dirichlet's Schubfachprinzip" (which apparently has something to do with "drawers").
Your second claim fails for infinitely many real numbers.  In fact,
there are (uncountably) infinitely many real numbers $x$ such that for any fixed $d$, no matter how large, there are infinitely many coprime integers $p,q$ with $|x-\frac{p}{q}| \leq \frac{1}{q^d}$.  The standard example is $\sum_{n=1}^{\infty} 10^{-n!}$.
I suppose that what you have in mind is a celebrated theorem of Klaus Roth from the 1950's.   What Roth proved -- greatly improving on previous results of Liouville, Thue, Siegel, Gelfond and Dyson -- is that this holds for all real algebraic numbers.
I give an introduction to this circle of ideas in $\S$ 14 of these lecture notes on elliptic curves.  Early on I include a proof of Dirichlet's Theorem together with some classical number theoretic applications: Pell's Equation.  I do not prove Roth's Theorem, although I state and use an even stronger theorem of Roth-Ridout.  For a detailed proof of Roth-Ridout I recommend the GTM Diophantine Geometry by Hindry and Silverman.
A: I think the first is comparatively easy, but the second refers to one of the landmark theorems of $20^{\rm th}$ century mathematics, the so-called Thue-Siegel-Roth theorem. Roth got a Fields medal for it. It's still a "solitaire" theorem; it has not become part of a larger theory. Note that the statement only holds for algebraic numbers. The number
$$\sum_{k=1}^\infty 10^{-k!}=0.11000100000000000000000100\ldots$$
(or similar) has infinitely many approximations of the described kind.
