Completeness property of R

Question

I was reading up on the completeness property of R that sets it apart from Q

It has been stated in 2 ways which are

1. Every non empty subset of R which is bounded above has a supremum in R

2. Every non empty subset of R which is bounded below has an infimum in R

Now this is very clear. The problem lies in what these 2 statements are followed by which has been marked in red in the given picture. It says that these 2 properties are equivalent in the sense that one implies the other. How is that even possible?

according to thus language, they r suggesting that 1. implies 2. i.e. if supremum exists than infimum exists too and vice versa which is clearly not the case. Am i interpreting it wrong?

Can someone help urgently plz!

Answer 1

Take a non-empty subset $S$ of $\mathbb{R}$ that is bounded below.

Let $T$ be the set $S$ where every element is multiplied by $(-1).$

Therefore, $T$ is bounded above.

By assumption, $T$ has a supremum.

Therefore, $S$ has an infinum.