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The full question given in the example is:

Let $X$ be a set and $d_0$ is the discrete metric for $X$. Suppose that $(a_k)$ is a sequence in $X$ that is $d_0$-convergent. Show that $(a_k)$ is an eventually constant sequence.

The solution is given as:

Let $(a_k)$ be a sequence in $X$ and suppose it converges (in $d$) to $a ∈ X$. Then by the definition of convergence, it must be the case that $(d_0(a_k,a))$ is a real null sequence and so, in particular, there is $N∈\mathbb{N}$ such that for $k > N$, $\lvert{d_0(a_k, a)}\lvert<1$.

But $d_0(a_k,a)$ can only equal $0$ or $1$, so this means that there must be $N∈\mathbb{N}$ such that for $k>N$, $d_0(a_k,a)=0$. Since $d_o$ is a metric on $X$, property M1 tells us that for $k>N, a_k=a$. In other words, the sequence $(a_k)$ is eventually constant, as required.

I don't understand why they state $\lvert{d_0(a_k, a)}\lvert<1$. Where does that come from? It anyone could clarify that for me it would be greatly appreciated!

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    You may pick any $\epsilon>0$. Picking $\epsilon=1$ leads to the desired conclusion. – 2017-02-18
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    How important is it that $ϵ=1$? If I took it to be 10, would the answer still hold? I feel like it wouldn't because taking it as less than 1 left the only possible outcome as 0, which was the crux of the explanation if my understanding is correct. – 2017-02-18

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Hint:

By definition:

a sequence $\{a_k\}$ has limit $a$ iff for each real number $\epsilon > 0$ there exists a number $N$ such that, for every natural number $n > N$ we have $d( a_n,a ) < ϵ$

Use $\epsilon=1$

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    Ahh I see. Thank you! – 2017-02-18
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    How important is it that ϵ=1? If I took it to be 10, would the answer still hold? I feel like it wouldn't because taking it as less than 1 left the only possible outcome as 0, which was the crux of the explanation if my understanding is correct. – 2017-02-18
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    The discrete metric has values in $\{0,1\}$, so any value $0<\epsilon\le 1$ works. – 2017-02-18
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    Yeah I thought so. I'm trying to understand this example, so that I can attempt the same question, but in the case of $d(x, y) = 5\lvert{x-y}\lvert + d_0(x, y)$. I don't think I can use the same argument on it, due to the $5\lvert{a_k-a}\lvert$ part of $d(a_k, a)$ giving other possible answers when I set $ϵ=1$. Do you have any pointers or hints in showing it in this case? – 2017-02-18