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@@ -4,7 +4,9 @@
 
 #<b>Why the Fourier decomposition is possible?</b>
 
-A Fourier series is a function of the form $\sum_{n=-\infty}^\infty a_n e^{inx}$ or $\sum_{n=-\infty}^\infty b_n \cos(n x) + c_n \sin (n x)$, depending on one's taste for the imaginary. The rumor on the street is that any periodic function (well, any nice one) can be expressed as a Fourier series: you hand me a function $f:[0,2\pi]\rightarrow \mathbb{C}$ and I hand you a list of real numbers $b_0,c_0,b_1,c_1,b_2,c_2,b_3,c_3,\dots$ such that
+A Fourier series is a function of the form 
+$$\sum_{n=-\infty}^\infty a_n e^{inx}$$ or $$\sum_{n=-\infty}^\infty b_n \cos(n x) + c_n \sin (n x),$$
+depending on one's taste for the imaginary. The rumor on the street is that any periodic function (well, any nice one) can be expressed as a Fourier series: you hand me a function $f:[0,2\pi]\rightarrow \mathbb{C}$ and I hand you a list of real numbers $b_0,c_0,b_1,c_1,b_2,c_2,b_3,c_3,\dots$ such that
 
 $$f(x) = \sum_{n=-\infty}^\infty b_n \cos(n x) + c_n \sin (n x).$$
 
@@ -12,9 +14,10 @@ If this exchange is always possible, there must be at least as many different li
 
 Infinity, of course. So to perform a meaningful check that the set of nice periodic functions is the same size as the set of lists of real numbers, we need a more refined notion of a size of a set than just giving its number of elements, or saying that it is infinite. We say that two sets have the same cardinality (our new word for size) if we can pair off their elements, giving a perfect matching between the sets.
 
+For example, the open interval $(0,1)$ has the same cardinality as $\mathbb R$. To see this, consider 
 
 ## To show that $(0,1) \sim \mathbb R$
-*--> could someone fill this out?  *
+
 
 ## Cantor's proof for $\mathbb R > \mathbb N$
 Suppose that we had a bijection between $\mathbb N$ and $\mathbb R$. Since $\mathbb R \sim \mathbb N$ In other words, we have a list of all the real numbers