When suspension and reduced suspension agree

Question

In the nLab entry, suspension, it says that for CW complexes, suspension and reduced suspension agree up to weak homotopy equivalence.

My questions are:

1. How to prove this result? Any reference would be okay for me.

2. Is there any intuitive counterexample to see suspension and reduced suspension can disagree?

Edit: I should be more clear about my questions. So I add some updates as follows.

For question 1: According to Example 0.10 of Hatcher's Algebraic Topology, the reduced suspension is defined for any CW complex with a fixed point $x_{0}$ as $0$-cell. So as Hatcher says, we can shrink the 1-cell $x_{0}\times I$, which is possible to write down the homotopy we need. So I admit this question is weird. I should ask what if we don't need a fixed point.

Okay, I just realized that we can make any point as a $0$-cell (for example, see this MSE question).

For question 2: I have tried some common spaces that are not CW complexes. I find imagining those shapes does not provide any clear clue. I would like to know a clear and intuitive example, or, a solid proof for a space.

Answer 1

1. The suspension $SX$ has $\{*\} \times I$ as a subcomplex. Since it is contractible, the map $SX \rightarrow \Sigma X$ from the unreduced to the reduced to the reduced suspension is a homotopy equivalence.

2. If $Z=\{0\} \cup \{1,\frac{1}{2},\frac{1}{3},\dots\}$ with basepoint 0, it is a fun exercise to see that $SZ$ is not homotopy equivalent to $\Sigma Z$.