This is an interactive website to help you learn how to write a sequence convergence 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 $$x_n := \frac{n^2-3}{n^2+1}$$. Prove that $$(x_n) \rightarrow 1$$.

Proof: We compute that $$x_n =\frac{n^2-3}{n^2+1}=\frac{1-\tfrac{3}{n^2}}{1+\tfrac{1}{n^2}}$$. Now $$\left(1-\tfrac{3}{n^2}\right)\rightarrow 1$$ because $$p(x)=1-3x^2$$ is a polynomial and $$\left(\tfrac{1}{n}\right)\rightarrow 0$$. Similarly $$\left(1+\tfrac{1}{n^2}\right)\rightarrow 1 \neq 0$$ because $$1+x^2$$ is a polynomial. Thus $$\left(\frac{1-\tfrac{3}{n^2}}{1+\tfrac{1}{n^2}}\right)\rightarrow \tfrac{1}{1}=1$$ by CoS (Combinations of Sequences).