To show that $u = \inf S$, we need to demonstrate two things:

1. $u$ is a lower bound of $S$.
2. $u$ is the greatest lower bound of $S$.

1. $u$ is a Lower Bound of $S$

Given:

• For every $n \in \mathbb{N}$, $u - \frac{1}{n}$ is a lower bound of $S$.

This means for all $s \in S$ and for each $n \in \mathbb{N}$, $u - \frac{1}{n} \leq s.$

Since this inequality holds for every $n \in \mathbb{N}$, taking the limit as $n \to \infty$: $\lim_{n \to \infty} \left( u - \frac{1}{n} \right) = u.$ So, $u \leq s$ for all $s \in S$. Therefore, $u$ is a lower bound of $S$.

2. $u$ is the Greatest Lower Bound of $S$

To show $u = \inf S$, we need to prove that $u$ is the greatest lower bound, i.e., if $l$ is any lower bound of $S$, then $l \leq u$.

Given:

• For every $n \in \mathbb{N}$, $u + \frac{1}{n}$ is not a lower bound of $S$.

This implies that for each $n \in \mathbb{N}$, there exists some $s_n \in S$ such that: $u + \frac{1}{n} > s_n.$

Since $s_n \in S$, and $u$ is a lower bound of $S$, we have $u \leq s_n$. Therefore, for each $n$: $u \leq s_n < u + \frac{1}{n}.$

Thus, $s_n - u < \frac{1}{n}.$

Taking the limit as $n \to \infty$: $\lim_{n \to \infty} (s_n - u) = 0,$ which means $\lim_{n \to \infty} s_n = u.$

Since for any $\epsilon > 0$, we can choose $n$ large enough such that $\frac{1}{n} < \epsilon$ and $u + \frac{1}{n}$ is not a lower bound, we find $s_n$ that is within $\epsilon$ of $u$. Therefore, $u$ is the greatest number such that $u$ is less than or equal to all elements of $S$.

Since $u$ is a lower bound and no number greater than $u$ can be a lower bound (because there exists $s_n \in S$ arbitrarily close to $u$ from above), $u$ is the greatest lower bound of $S$.

Hence, $u = \inf S$.

Would you like to see more details on any specific part of this proof, or do you have any other questions?

Here are some related questions for further exploration:

1. What are the implications of a set having an infimum in the real numbers?
2. How would the proof change if we considered the supremum instead of the infimum?
3. Can $u$ be an element of $S$? If so, under what conditions?
4. How would this proof differ if $S$ were a subset of $\mathbb{Q}$ (rationals) instead of $\mathbb{R}$?
5. Can we use this method to approximate the infimum of a set $S$ numerically?

Tip: Remember, the infimum of a set $S \subseteq \mathbb{R}$ might not be an element of $S$, but it is the greatest lower bound that $S$ has in the real numbers.