Prove that $a_n$ is null sequence for $a_{n}=n$ when $10\ge n$ and $a_{n}=\frac1n$ when $n>10. $

Question

I don't want the proof for this question since I have already done one. But what I want is the explanation for the counterintuitive proof that the book provided me with.Here is it:

Let me give the details about 36(a) and 36(c).

36(a) Every subsequence of null sequence is null.

36(c) For some fixed positive integer k, The sequence $a_{n+k}$ is null if $a_{n}$ is null.

Explain me the proof in photo not the definition 36 (a and c).

Thanks for entertainment!