So I am trying to solve questions below
Let $A = \{(\alpha_1,\alpha_2,\alpha_3,\ldots): \alpha_i \in \{0,1\}, i \in N\}$, i.e., $A$ is the infinite cartesian product of the set $\{0,1\}$. Show that $A$ is uncountable.
Prove that the intervals $[0,\infty)$ and $(-1,4)$ have the same cardinality by constructing a bijection between the two sets.
For the second question I tried to construct a bijection between $[0,\infty)$ and $[-1,4)$, but couldn't go any farther.
Thank you so much for any help!
For the first, this is a standard example for cantors diagonal counting,
the second one would be much easier with Cantor Schröder Bernstein. If you want a bijection we construct at first a bijection from $[0,\infty)\to (0,\infty)$, use the identity on every non integer and for the integers add $1$. After this we can use the function $\frac{1}{x+1}$ to map $(0,\infty)$ on the set $(0,1)$. Finding a bijection between to open sets is now quite easy so you should be able to handle the rest.