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I want to prove that number of points in a plane is equal to the number of points on $ [0, 1]$.

I think of assuming a function $ f\colon \mathbb{R} \to \mathbb{R}\times\mathbb{R} $ such for every $x$, $f (x)=(x, 0) $. Is this right? May I say that $ \mathbb{R} $ and $ \mathbb{R}\times\mathbb{R} $ have the same number of points so $ \mathbb{R}\times\mathbb{R} $ has the same number of points as $[0, 1] $ does?

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    $\begingroup$ It's important to use correct English : "I wanna" $\rightarrow$ "I want to." Besides it has just been corrected by $Matthew Leingang $\endgroup$
    – Jean Marie
    Sep 5, 2016 at 18:26
  • $\begingroup$ Your function is only an injection from $\mathbb{R}$ to $\mathbb{R}\times \mathbb{R}$ so you've only proven $|\mathbb{R}|\leq |\mathbb{R}\times\mathbb{R}|$. Take a look at, math.stackexchange.com/questions/1291677/… $\endgroup$
    – Nap D. Lover
    Sep 5, 2016 at 18:34

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