Describing sequences that satisfy $\forall \delta>0 : \forall N \in \mathbb{N} : \exists n>N, |b_n-L|<\delta$ i have been thinking about what would happen if i changed some of the quantifiers in the definition of convergence of a sequence.
i am wondering what property (or properties) (general form/description) sequences satisfying
$\forall \delta>0 : \forall N \in \mathbb{N} : \exists n>N, |b_n-L|<\delta$
have, where $b_n$ is the $n^{th}$term of the sequence and $L$ is the limit of the sequence as $n$ tends to infinity.
I am not sure how to approach this and would appreciate some help.
 A: Your condition implies (and is equivalent to) the sequence having a subsequence convergent to $L.$ For example, if your sequence is that of the rationals in some order, you will get that this property will be satisfied for any $L.$
A: In English, this property is "an infinite subsequence of $(b_n)$ converges to $L$".
If an infinite subsequence $(b_{m_i})$ converges to $L$, the definition of convergence means for any $\delta>0$ there is some $M$ such that $i>M$ implies $|b_{m_i} - L| < \delta$, so for any $N$ we can take $K = \max(M, N) + 1$ and $n=m_K>N$ identifies an element $b_n$ satisfying $|b_n - L| < \delta$. Your property is also satisfied.
If $(b_n)$ satisfies your property, then for any $\gamma > 0$ we can take $N=1$ and $\delta=1$ to find $m_0$ with $|b_{m_0}| < 1$; then recursively take $N=m_n$ and $\delta=2^{-n}$ to find $m_{n+1}$ with $|b_{m_{n+1}} - L| < 2^{-n}$. Then $(b_{m_i})$ is an infinite subsequence which converges to $L$.
A: If a sequence $(b_n)_n$ satisfies, for $L \in \mathbb R$ (let's work with $\mathbb R$ for the sake of simplicity),
$$\forall \delta>0 : \forall N \in \mathbb{N} : \exists n>N, |b_n-L|<\delta$$
then $(b_n)_n$ admits a subsequence converging toward $L$. Indeed, define $(b_{n_k})_k$ the following way:
$$n_k = \min \{s > n_{k-1}~|~|b_s- L| < 1/k\}.$$
You may check that $(b_{n_k})_k$ converges towards $L$.
