If $(x'_n)$ is Cauchy in the quotient space $X/Y$, then the corresponding sequence $(x_n)$ may not be Cauchy in $X$

Question

If $(x'_n)$ is a Cauchy Sequence in the quotient space $X/Y$, then the corresponding sequence $(x_n)$ may not be a Cauchy Sequence in the normed space $X$.

I tried to find an example of such kind of sequence $(x'_n)$ in some quotient space $X/Y$ (I took $R^3/(x,0,0)$ etc.), but could not find. Kindly give some example of such $(x'_n)$ in any quotient space.

Answer 1

Let $X = \mathbb{R}^2$ and $Y = \{(0,y) \mid y \in \mathbb{R}\}$. It's not hard to see that $X/Y$ is isomorphic to $\mathbb{R}$. Let $p : X \to X/Y$ denote the quotient map.

Now suppose $(x_n)$ is Cauchy in $\mathbb{R}$ and observe that the the sequence $((x_n,n))$ is not Cauchy in $\mathbb{R}^2$. However, $p(x_n,n) = x_n$ and we know that $(x_n)$ is Cauchy.

Answer 2

Hint: let $(x_n)$ be any sequence in $Y$. Then $x_n'=0'$ for all $n$.