This is an interactive website to help you learn how to write an \(\varepsilon - K\) proof. You should try to prove the statement yourself, and use the information on this webpage to check your work or to get a hint if you need one.

**You can see that most of the proof is missing.**
Click on the boxes to reveal the missing expressions.

But * before you click*, try to write the proof yourself and figure out what will appear!

If you wish to make the text disappear again, refresh the page.

### Let \(x_n := \frac{7}{n}\). Prove that \((x_n) \rightarrow 0\).

*Proof:* Given \(\varepsilon >0\), let \(K > \tfrac{7}{\varepsilon}\). Then if
\(n \ge K\) we have
\(|x_n - 0| = \)
\(\left|\tfrac{7}{n}\right|=\)
\(\tfrac{7}{n}\le \)
\(\tfrac{7}{K}< \)
\(\tfrac{7}{7/\varepsilon}=\)
\(\varepsilon\).
Thus \(\left(\tfrac{7}{n}\right)\rightarrow 0\) by the Definition of Sequence Converence.

Move on to Example 2