Bijection between finite and infinite sequences over Reals.

Question

So define the set of finite sequences to be $S={a_1,a_2,\cdots}$ where $a_k$ are in real numbers and only finitely many of them are non-zero. The set of infinite sequences is defined similarly except that we can have infinitely many non-zero terms. How do I prove that there does not exist a bijection between these two sets?

Answer 1

You don't, because there exist such bijections. The first one is (essentially) $\displaystyle\bigcup_{n\in \mathbb{N}} \mathbb{R}^n$, which has the same cardinality as $\mathbb{R}$, and the second one is $\mathbb{R}^\mathbb{N}$, which also has the same cardinality as $\mathbb{R}$.