Definition: A tuple $\lambda = (\lambda_1, \ldots, \lambda_k)$ of natural numbers is called a numeric partition of $n$ if $1 \leq \lambda_1 \leq \cdots \leq \lambda_k$ and $\lambda_1 + \cdots + \lambda_k = n$ and is written as $\lambda \vdash n$.
Exercise: Let $p(n)$ be the amount of numeric partitions of $n$. Prove that
i) Let $f_n := \prod_{i=1}^n (1-x^i)^{-1}$. Prove that $(f_n)_{n\geq 0}$ creates a Cauchy sequence in $\mathbb{C}[[x]]$.
ii) Prove that $ \sum\limits_{n \geq 0} p(n) x^n = \lim\limits_{n \rightarrow \infty} f_n = \prod\limits_{i=1}^\infty (1-x^i)^{-1} .$
I tried to play around with the definition of $f_n$ and tried (quite successfully as I hope) to understand what a Cauchy sequence is. But how do I prove that?
What should I do to prove the second one? Any hints?
Thanks in advance!