1. ## sujective function

Is there a surjective function from R^m to R^n where m<n ?

My guess is no but I don't know how to justify it.

Thansk

2. No surjective function from $\displaystyle \mathbb{R}^{m}$ to $\displaystyle \mathbb{R}^{n}$ with $\displaystyle m<n$?

Actually, for all $\displaystyle m$ and $\displaystyle n$ in $\displaystyle \mathbb{N}-\{0\}$, $\displaystyle \mathbb{R}^{m}$ and $\displaystyle \mathbb{R}^{n}$ are equipotent (i.e. there exists a bijective function between them)!

Of course, if $\displaystyle m=0$, $\displaystyle \mathbb{R}^{0}$ is finite while $\displaystyle \mathbb{R}^{n}$ is infinite, so there is no surjection.

Let $\displaystyle A$ and $\displaystyle B$ be two sets, Cantor-Shröder-Bernstein theorem states that if there exists an injection from $\displaystyle A$ to $\displaystyle B$ and an injection from $\displaystyle B$ to $\displaystyle A$, then there exists a bijection between them.

So, to start, you can find an injective function from $\displaystyle \mathbb{R}$ to $\displaystyle ]0,1[$, to prove they are equipotent, then do the same for $\displaystyle \mathbb{R}^{2}$ and $\displaystyle ]0,1[^{2}$, and finaly find a surjective function from $\displaystyle ]0,1[$ to $\displaystyle ]0,1[^{2}$, so that will prove there is a surjection from $\displaystyle \mathbb{R}$ to $\displaystyle \mathbb{R}^{2}$.

But there can be a quicker way...