📘 real analysis

1. The problem asks to find the supremum (least upper bound) and infimum (greatest lower bound) of the set $$S=\left\{\frac{n-m}{n+m}: n,m \in \mathbb{N}\right\}$$ where $n$ and $m

1. The problem is to show that the set $ (1,2) $ has no maximum element. 2. The set $ (1,2) $ is an open interval, meaning it includes all real numbers between 1 and 2 but does not

1. First, state the inequalities to prove: a. For all real numbers $x, y$, prove that $|x| + |y| \leq |x + y| + |x - y|$.

1. **Show the inequalities in Exercise 2:** **a.** Show that $|x|+|y| \leq |x+y|+|x-y|$ for all $x,y \in \mathbb{R}$.