Difficulty(3) in understanding the completeness of $l_{2}.$

Question

Here is a part of the proof:

why the author put the equality on the inequality after taking $n \rightarrow \infty$, could anyone clarify this for me please?

Answer 1

Recall that the process of taking limits does not in general preserve strict inequalities. For example consider $\frac{1}{n} > 0$ for any $n \in \mathbb{N}$ but we clearly have equality in the limit $n \to \infty$.

Taking limits does however preserve weak inequalities which is enough to give the authors result.

Answer 2

This is standard when working with inequalities and limits. If $a_{n}<b_{n}$ and $lim a_{n}=a$ and $lim b_{n}=b$ then the most we can say is $a \leq b$. For example take $a_{n}=0$ and $b_{n}:=1/n$. We know that $0<1/n$, but the limits of both sides are equal.

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Problems with the proof that $\ell^p$ is complete

Which contains a very smart explanation for completeness of $l_{p}$.