What does the set of reals with infinitely many sequence terms below it represent?

[sonofagun]

I'm reading the proof that a cauchy sequence is convergent.

Let a_n be a cauchy sequence and let ε=1. Then ∃N∈ℕ such that for all m, n≥N we have
a_n-a_m<1. Hence, for all n≥N we have a_n-a_N<1 which implies a_n<a_N+1. Therefore, the set {n∈ℕ: an≤a_N+1} is infinite and thus {x∈ℝ : {n∈ℕ: a_n≤x} is infinite} ≠ ∅.

I can't make sense of the last set. What does it represent and why is it not empty?

 

[statdad]

Do you buy the statement that $\{n \in \mathbb{N} \colon a_n \le a_N + 1 \}$ is infinite? If so, what does that tell you about the $a_n$ themselves?

 

[sonofagun]

Do you buy the statement that $\{n \in \mathbb{N} \colon a_n \le a_N + 1 \}$ is infinite? If so, what does that tell you about the $a_n$ themselves?

Okay, I get it now. It's the set of all reals that are ≥ a_n for infinitely many n's. a_N+1 is an element in this set, thus it's not empty.
I'm studying independently so I occasionally get stuck trying to figure out easy concepts like this. Thank you!