Knowing the definition of convergence of a sequence in $\Bbb R$: ($x_n$) converges to x iff $\forall \epsilon \gt 0$ $\exists N$ such that for every $n\gt N$, $d(x_n,x)\lt \epsilon$.
Consider a new definition, say $C-convergence$: ($x_n$) C-converges to x iff $\exists N$ such that for every $\epsilon \gt 0$ and every $n\gt N$, $d(x_n,x)\lt \epsilon$.
Does C-convergence imply convergence? Also, does convergence imply C-convergence? Could someone provide a proof or counterexample to solve this problem? I'm confused about switching the order of the phrases in this definition. Thanks.