In $(]0,1], d_{\mid.\mid})$ why $(\frac{1}{1+n})_{n\in \mathbb{N}}$ do not converge?

Question

Can someone tell me why this sequence do not converge ?

First, I know that is a Cauchy's sequence.

Then, the fact is that the sequence converges to $0$ when $n \rightarrow \infty$.

Thanks in advance.

Answer 1

This sequence would converge if our space of reference contained $0$. But as our space: $]0,1[$ DOES NOT contain $0$, then the sequence $\{\frac{1}{n+1}\}_{n\in\mathbb N}$ DOES NOT converge in $]0,1[$.

Clearly, the same sequence DOES converge in $[0,1]$, as $0\in [0,1]$!

Answer 2

As you say the sequence converges to $0$ when is seen as a sequence in, say, the real line $\Bbb R$.

But since you are looking at the open interval $(0,1)$ which does not contain $0$, the sequence is not convergent there. Quite simply.