Does an increasing sequence always have a lower bound?

Question

In the proof that I am trying to understand, it says: let $a(n)$ be increasing, then $a(1)$ is less than or equal to $a(n)$ for all $n$, so the sequence is bounded below. Does that mean that an increasing sequence is always bounded below?

Answer 1

Yes. Essentially a sequence being increasing means

$$a_1\leq a_2\leq a_3\leq a_4\ldots$$

Its clear that $\forall n\in \Bbb N, a_1\leq a_n$, hence $a_1$ is a lower bound.