# Thread: sujective function

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 $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$ with $m?

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

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

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

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

But there can be a quicker way...