Prove that the set of all finite sequences of real numbers has the same cardinality as the set $\mathbb{R}$ of reals. Prove that the set of all finite sequences of real numbers has the same cardinality as the set $\mathbb{R}$ of reals.
I can not understand the purpose of the question.

Let $A\sim B$ mean $A,B$ have the same cardinality. You want to show $\cup_{n=1}^\infty\mathbb {R}^n \sim \mathbb {R}.$ Assuming you know that each $\mathbb {R}^n\sim \mathbb {R},$ we can do this: There are bijections from ${R}^n$ to $[n,n+1)$ for each $n$ (because $[n,n+1)\sim \mathbb {R}),$ hence there is a bijection from $\cup_{n=1}^\infty\mathbb {R}^n$ to $[1,\infty) \sim \mathbb {R}.$
