I'm trying to understand the following examples:
Let T be a sequence such that $ T = \{1 - \frac{1}{n}\}$
It seems obvious that $\sup(T) = 1$
My proof for the supremum was:
For any $u < 1$ we have $\epsilon = 1 - u > 0$
By the Archimedean property there exists an $n \in \mathbb{N}$ such that $\frac{1}{n} < 1 - u$.
Or equivalently, $u < 1 - \frac{1}{n}$. Thus u is not an upper bound.
But I'm having trouble determining and proving $\inf(T)$. Could someone help me out?
Also, I'm completely lost on this example:
Let K = $[-\pi,\pi] \cap \mathbb{Q}$
I said that this is equivalent to saying:
K = $ \{k \in \mathbb{Q} : -\pi \leq k \leq \pi \}$
So from there I said $\inf(K) = -\pi$ and $\sup(K) = \pi$, but I'm not sure on how to go about proving them.
Thanks in advance.