$[Real Analysis] Show that $B$ is countable.$

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$[Real Analysis] Show that $B$ is countable.$

Let

$B = \{f :{\displaystyle \mathbb {R}}→{\displaystyle \mathbb {R}}$ such that $f(x) =$$\sum_{n=1}^{N} a_nx^n$, $N ∈{\displaystyle \mathbb {N}}, a_1, . . . , a_N ∈{\displaystyle \mathbb {N}}\}$.

Show that $B$ is countable.

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Let for all $N\in\mathbb{N}$
$A_N = \left\{(a_1,\dots, a_N)\in \mathbb{R}^N\right\}$.
Each $A_N$ is countable, since they're indexed by a finite set. This implies that
$ \bigcup_{N=1}^\infty A_N = \{(a_1, \dots, a_N): a_i\in\mathbb{R} \text{ for all } i\} $
is countable too.

Since $x$ is arbitrary, each function $f\in B$ is uniquely determined by its $(a_n)_{n=1}^N$ coefficients, thus we can find a bijection $\zeta: B \to \bigcup_{N=1}^\infty A_N$ by
$\sum_{n=1}^N a_n x^n \overset{\zeta}{\mapsto} (a_1,\dots, a_N)$.

Hence we can conclude that $B$ is countable.

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$[Real Analysis] Show that $B$ is countable.$

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