Let $(x_k)_{k∈\mathbb{N}}$ ⊂ $\mathbb{R^d}$ and $y$ ∈ $\mathbb{R^d}$. Show that $$(1)\ \ \ x_k → y ⇐⇒x_k-y→ (0,...,0) $$ And $$(2)\ \ \ x_k → y ⇐⇒||x_k-y||_2→ 0$$
$$$$ For (1) I started like this: Let $ε>0$ and also let $ε>y$ so that $ε-y>0$ for any y big enough so that $ε-y$ can be relatively small to 0. And there exists an $M>0$: if $k>M$ then $|x_k-y|< ε-y$. Now $x_k< ε-y+y ⇒x_k<ε$, here since $ε$ is just a limit is proven that $x_n$ converges, and since $x_n$ is a sequence, the limit of the sequence will be a sequence of limits, in our case a sequence of zeros.$$$$I don't know if this is the correct proof because I'm just a beginner in these topics, so if this isn't the right answer please could you help me with a proper solution for both subtasks?