Absolute convergence implies convergence.

Question

1. Here (https://mathcs.org/analysis/reals/numser/proofs/absconv.html) we read that absolute convergence implies convergence. I understant the proof and I am glad. Also there is an example which shows that converse is not true in general.

Okey.

2. In this (http://www.math.usm.edu/lee/mat771/hw3sol.pdf) paper on page 5, section 5 we read that in normed space convergence implies absolute convergence. ?!!!.

Question 1. About what space did we talk in 1?

Question 2. In the text bellow (from second link) we see that in normed space convergence implies absolut convergence.

But in the first link there is example when convergent sequence does not implies absolute convergence.

How to understand it? In this example we talk about R. And it is complate.

EDIT: Question 3. In wikipedia there is written that for absolute convergence we need norm. So we talk about absolute convergence only in the norm space. How to link this and the example above where we see that convergence does not implies absolute convergence?