Is this Proof for an Infinite Limit Correct?

[Mathematicsresear]

<Moderator's note: Moved from a technical forum and thus no template.>

1. Homework Statement
Is this proof correct?
Let K>0, and choose N such that N >= K², then for all n in the naturals, and n>=N, sqrt(n)+7>=sqrt(N)>=K

Is this proof correct?

Please tell me

 

[RPinPA]

Yes, you need to wrap a little formal language around it to make it a formal proof, but that's the argument. You might want to state and prove that $\sqrt{n}$ is monotonically strictly increasing, i.e. $\sqrt{m} > \sqrt{n} \iff m > n$, unless that's already been established in your course.

So I'd wind up with a closing line, "therefore [state the definition of going to $\infty$ as $n \rightarrow \infty$]".