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

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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?