I have these two definitions for an adherent value of a sequence
the first is : $a$ is a an adherent value for $(x_n)$ iff $$\displaystyle \forall \varepsilon>0,\forall n\in \mathbb{N},\exists n_0\geq n, d(x_{n_0},a)\leq \varepsilon$$ the second is $a$ is an adherent value for $(x_n)$ if there exists a sub sequence witch converge to $a$
But i don't know how to prove the equivalence between the two definitions
If there exists a subsequence which converges to $a$, it's clear the first definition is satisfied. Conversely, suppose the first definition is satisfied. We can define a subsequence that converges to $a$ in the following way: start with $\varepsilon=1$, and let $i_0$ be the first subscript $i$ such that $\lvert x_i-a\rvert < \varepsilon$.Then set $ \varepsilon:=\dfrac\varepsilon{10}$, and let $i_1$ be the first subscript $i>i_0$ such that $\lvert x_i-a\rvert < \varepsilon$. This defines recursively a subsequence $(v_k)=(u_{i_k})$ of $(u_i)$ which converges to $a$ by construction.
