I want to prove that the closed interval [0, 1] has same cardinality as Real Numbers. I was able to figure out (0, 1), but need help with proving the closed part. What I have so far is:
the function $y = tan x$ gives us a bijection between $(−π/2, π/2)$ and $R$, and that the function $y = (2x − 1)π/2$ gives a bijection between $(0, 1)$ and $(−π/2, π/2)$. Therefore $y = tan (2x − 1)π/2$ gives a bijection between (0, 1) and $R$.
How do I finish this off for the closed aspect?