# What does this set represent?

1. Mar 5, 2015

### sonofagun

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

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

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

2. Mar 5, 2015

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?

3. Mar 5, 2015

### sonofagun

Okay, I get it now. It's the set of all reals that are ≥ an for infinitely many n's. aN+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!