An exercise on uncountable subsets of $[0,1]$

Question

I am stuck on how to prove these three questions, or even how to draw the sets so I can see where they overlap. Any help would be appreciated!

1. Let $C\subseteq [0,1]$ be uncountable. Show there exists a in $(0,1)$ such that $C\cap[a,1]$ is uncountable.
2. Let $A$ be the set of all $a \in (0,1)$ such that $C\cap[a,1]$ is uncountable, and set $p=\sup A$ Is $C\cap[p,1]$ an uncountable set?
3. Does question 1. remain true if "uncountable" is replaced by "infinite"?

Answer 1

Hint for 1: By contradiction. Start by observing that $C\setminus\{0\} = \bigcup_{n=1}^\infty \left( \left[\frac{1}{n},1\right]\cap C\right)$ (and $C\setminus\{0\}$ is uncountable).

Hint for 2: By contradiction, suppose $[p, 1]\cap C$ is uncountable. Can you apply the above argument (i.e., question 1) to $C^\prime = [p, 1]\cap C$ and $I=[p,1]$, instead of $C$ and $[0,1]$?

Hint for 3: take $C= \{\frac{1}{n}\mid n \in \mathbb{N}^\ast\}$.

Answer 2

1) If for every $a = \frac{1}{n} \in (0,1)$, where $n \in \mathbb N$ we would have $C \cap \left[\frac{1}{n},1\right]$ countable then union $$\bigcup_{n=1}^{\infty} C \cap [1/n, 1] = C \cap \bigcup_{n=1}^{\infty}[1/n, 1] = C \cap [0,1] = C-\{0\}$$

would be countable, which is a contradiction.