Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I learned the following: $\forall n \exists k_0 : \forall k\ge k_0: |a_k - a^*|<n$. And my textbook also pointed that you can pick some n, let's say $n=1/k$. But that's my struggle; then you can choose anything for n? But then there is always a way for the equation to be true? And so every sequence must have a limit?

Regards, Kevin

share|cite|improve this question
up vote 5 down vote accepted

First of all, the use of $n$ is quite confusing since usually it denotes something going to infinity, for infinitely small values one usually use $\varepsilon$. See e.g. P. Halmos:

A mathematician’s nightmare is a sequence $n_ε$ that tends to $0$ as $ε$ becomes infinite.

So, the definition of $$ \lim\limits_{k\to\infty}a_k = a^* $$ is: for any $\varepsilon>0$ there exists $k_0(\varepsilon)$ such that for all $k>k_0(\varepsilon)$ it holds that $|a_k-a^*|<\varepsilon$.

So, formally you need to check that the condition above holds for all $\varepsilon>0$. Fortunately, it is equivalent to check for only sequence of $\varepsilon$ which is positive and converge to zero itself (quite recursive, though). Clearly, $\varepsilon_i = \frac1i$ is one of the examples.

I hope I understand you post correctly, because $n=\frac1k$ leads to a question which $k$ do you mean?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.