Here is a question (See question $9$).
It says that we have the ring $K[[x,y]]$ and the ideal $I=\left\langle x,y \right\rangle$, where $K$ is a field.
We to show that the sequence $\{s_n=\sum_{i=0}^n r_ix^iy^{i+2}\}_{n \in \mathbb{N}\cup\{0\}}$ is a Cauchy sequence in the $I$-adic topology.
The same worksheet has the definition of Cauchy sequence.
The terms of the sequence are $$\{s_0=r_0y^2,~s_1=r_0y^2+r_1xy^3,~s_2=r_0y^2+r_1xy^3+r_2x^2y^4,~\cdots\}$$
we have to show that for all $t \in \mathbb{N}$, there exists $d \in \mathbb{N}$ such that whenever $n,m \geq d$ $$s_n-s_m \in I^t.$$ For $t=1$, $s_n-s_m \in I$.
For $t=2$, and $d=2$, whenever $n,m \geq 2$ we have $$s_n-s_m \in I^2=\left\langle x,y \right\rangle^2.$$
For $t=n$, we have $s_{n+1}-s_n \in I^{n+2}$.
So the general case is $$s_n-s_m \in I^{n-m+2}.$$
Hence the given sequence is a Cauchy sequence in the $I$-adic topology, I think.
Am I Correct?
In this regard, I have another question for local ring:
Question 2:
Let $O$ be a local ring with unique maximal ideal $m$. Let $\{f_n\}$ be a sequence of monic irreducible polynomials in $O[x]$ of degree $n$ having roots in $\bar{m}$. Let us define $F_n(x)=\prod_{i=1}^nf_n(x)$ and take direct limit $F=\lim_{n \to \infty} F_n$.
When is the sequence $\{F_n\}_{n\in\mathbb{N}}$ a Cauchy sequence in $m$-adic topology ?