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For this problem, let there is a sequence $\{y_n\}$ which is complete in $Y$ such that this sequence has finitely many distinct points and this sequence approaches to say $y$. In this case $y$ is repeated infinitely many times. In this scenario, there may exist open ball $B(y;r)$ which only contains $y$ and so $y$ can't be the accumulation point of this sequence. So however $Y$ is complete, it is not closed.

I am not able to understand the mistake.

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