Set Theory – Binary Sequences and 2^N Have the Same Cardinality

cardinalselementary-set-theoryfunctions

I recently got the book "selected problems in real analysis", and I'm stuck solving the very first problem

$(u_n)$ is a binary sequence iff it only contains $0$ and $1$ in the sequence

Let $A$ be the set of all binary sequences

I have to prove that $A$ and ${2}^{\mathbb{N}}$ have the same cardinality, that is to say there exists a 1-1 function from one set to another

I've thought about maybe considering integers as base-2 numbers

Thanks for your help

Best Answer

Hint: By $2^{\Bbb N}$ I presume you mean the power set of $\Bbb N$, rather than the set of functions $\Bbb N\to\mathbf{2},$ where $\mathbf{2}:=\{0,1\}$, as these are precisely the binary sequences, and there's nothing to prove. Have you heard of indicator functions?

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