Is this an increasing sequence?

Question

From this beginning of this document -

Let $(c_n)$ be a bounded sequence of real numbers.

Define $a_n = \inf\{c_k : k ≥ n\}$

The sequence $(a_n)$ is bounded and increasing so it has a limit $a$.

Question

This does not seem to be true to me as $(c_n)$ could be a constant sequence and hence $a_{n+1} = a_n$ for all $n$. I.e. $a_{n+1} > a_n$ for all $n$ is false. So is the document incorrect?

Answer 1

I think GitGud is right.

Consider JC Burkill, A First Course in Mathematical Analysis (1962), page 31 :

Definition. If $s_{n+1} \ge s_n$ for all values of $n$ we call $s_n$ increasing.

It is useful to regard increase in the wide sense, allowing the possibility of equality at any of the steps from $n$ to $n+1$. If $s_{n+1} > s_n$ for all $n$, we call $s_n$ strictly increasing.

Then see page 32 :

A bounded increasing sequence tends to a limit.