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**Pinkk** Okay, so I have a problem where $\displaystyle a_{n}=\frac{1}{n}$ and I need to show using the formal definition that $\displaystyle a_{n}$ converges to 0:

$\displaystyle \lim_{n\rightarrow \infty}a_{n}=L=0$.

So I let $\displaystyle |a_{n}-L| < \varepsilon$. Simplifying it, I get

$\displaystyle n > \frac{1}{\varepsilon + L}$, essentially getting within distance $\displaystyle \varepsilon$ of $\displaystyle L$ for any integer $\displaystyle N$ where $\displaystyle n > N$.

So then I get $\displaystyle n > N \ge \frac{1}{\varepsilon + L}$.

Doing some manipulation, I eventually get to $\displaystyle \varepsilon > \varepsilon\frac{N}{n}+L\frac{N-n}{n} \ge \frac{1}{n}$ This is where I get lost on showing that $\displaystyle L=0$.

Now my professor didn't really sufficiently explain how to do a formal proof even when we already assume we know the value of $\displaystyle L$. For instance, we know $\displaystyle L=0$ for this $\displaystyle a_{n}$. Knowing this, the inequality would become $\displaystyle \varepsilon > \varepsilon\frac{N}{n} \ge \frac{1}{n}$. Do we know that this is true because $\displaystyle \varepsilon\frac{N}{n}$ is always less than $\displaystyle \varepsilon$ since $\displaystyle \frac{N}{n} < 1$, or is there more that needs to be shown? If that is sufficient, than how can that line of thought be applied when we assume we don't know the value of $\displaystyle L$ (like in this problem)?

Sorry if there is already a thread that asks and explains this. Our class pretty much went through sequences today in about 30 minutes so I'm not too sure on this stuff. Thanks!