What is $\left(\bigcup\limits_{n=1}^\infty E_n\right)^c$? In an experiment, die is rolled continually until a 6 appears, at which point the experiment stops.

*

*What is the sample space of this experiment?

The sample space consists of sequence where the last entry must be equal to six and no other entry can be six. Mathematically, we can describe the space as
$$ S = \{(6), (x_1,6), (x_1,x_2,6), \dots, (x_1,x_2,\dots,x_i,6) | \text{ where } x_i=\{1,2,3,4,5\} \text{ and } i\geq 1\}$$

*

*Let $E_n$ denote the event that $n$ rolls are necessary to complete the experiment. What points of the sample space are contained in $E_n$?

$E_n$ contains all sequences in the sample space of length $n$. Mathematically, we can $E_n$ as
$$ E_n=\{(x_1,\dots,x_{n-1},6) | \text{ where } x_i=\{1,2,3,4,5\} \} $$

*

*What is $\left(\bigcup\limits_{n=1}^\infty E_n\right)^c$?

By De Morgan's Law,
$$
\left(\bigcup\limits_{n=1}^\infty E_n\right)^c = \bigcap\limits_{n=1}^\infty E_n^c
$$
So here is my question: Is $E_n^c$ is the set of $n$ length sequences where the last entry is anything but 6?
I am having a hard time believing that because then those sequences would not belong in the sample space $S$?
 A: Your sample space isn't complete given how you've defined the process it describes. We need to allow for arbitrarily long sequences of "non-sixes".
We can express this as the following
$$S:= \left(\bigcup_{i=1}^{\infty}\{1,2,3,4,5\}^i \times \{6\}\right)\cup \{6\}$$
Given above:
$E_n:\{1,2,3,4,5\}^{n-1} \times \{6\}, n>1$ and $E_1 = \{6\}$
If we start with just $E_1$ we get:
$$E_1^c = S\setminus \{6\} = \left(\bigcup_{i=1}^{\infty}\{1,2,3,4,5\}^i \times \{6\}\right)$$
$$\left(E_1 \cup E_2\right)^c =S \setminus \{\{6\} \cup \{1,2,3,4,5\}\times \{6\}\} =  \left(\bigcup_{i=2}^{\infty}\{1,2,3,4,5\}^i \times \{6\}\right)$$
In general,
$$S_n:=\left(\bigcup_{i=1}^n E_i\right)^c = \left(\bigcup_{i=n}^{\infty}\{1,2,3,4,5\}^i \times \{6\}\right),\;\;\forall n>0$$
We can see that the sequence of sets $S_i$ is strictly decreasing:
$$S_j \supset S_{j+1}\;\forall j>0$$
Therefore,
$$\lim_{n\to \infty} S_n = \{1,2,3,4,5\}^{\mathbb{N}} \times \{6\} =  \{1,2,3,4,5\}^{\mathbb{N}}$$ up to isomorphism
If this seems weird I agree but you can totally do this — we just define $6$ to be at the special index $\infty$ of our sequence.
See here: https://math.stackexchange.com/a/979217/632875
A: The $E_n$'s partition the sample space. For each $n\in\mathbb{N}$, $E_n$ is the set of $n$-tuples in $S$. Thus, $E_n^c$ is the set of $m$-tuples in $S$ where $m\ne n$. Also using this line of reasoning, $$\left(\bigcup_{n=1}^\infty E_n\right)^c=S^c=\emptyset$$
I am assuming that the sequences are finite and the experiment ends after finitely many steps.
