I'm reading Bartle's text (Real Analysis 4th edition), pg 42.
The following proof seems incomplete. The conclusion that he makes in the last sentence seems to be a leap of logic.
For example, how does he know that $1/n_t$ is actually less than $t$? Couldn't it be greater than $t$?
On a different note, is this a good book to use to learn this subject? I am new to proofs and real analysis.
On
If you're new to proofs, you might want to try:
Introductory Mathematics: Algebra and Analysis (by Geoff Smith)
The proof is complete as it is. This is a matter of quantification. If you were to pick an arbitrary number $n \in \mathbb N$, then of course you cannot say that $1/n < t$. However, this proof shows that there exists an $n_t \in \mathbb N$ such that $1/n_t < t$. If you take another look at the definition of lower bounds, this should become clear.
Becuase notice that $0$ is the infimum which is the greatest lower bound, meaning that any number greater than $0$, say $t >0$ for example ${\bf cannot}$ be a lower bound so there is some value in the set $\{ 1/n : n \in \mathbb{N} \}$ that is between $0$ and $t$, in this case $n_t$ so that
$$ 0 < \frac{1}{n_t} < t $$