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Let $V_1,V_2$ be two finite dimensional vector spaces with $\Bbb R$ as the scalar field. Under an arbitrary norm $\left\| \right\|$ defined on the product space $V_1 \times V_2$ (and let the norm on $V_1$ be induced as $\left\|x\right\|=\left\|(x,0)\right\|$, and likewise for $V_2$), will the convergence of $(x_k,y_k) \to (x,y)$ imply the convergences of $x_k \to x$,$y_k \to y$?

As $(x_k,y_k) \to (x,y)$ means for any $\epsilon >0$ there exists $N$ s.t. $k>N$ implies $\left\| {({x_k} - x,{y_k} - y)} \right\| = \left\| {({x_k} - x,0) + (0,{y_k} - y)} \right\| < \varepsilon $, but I can't see why $\left\| {{x_k} - x} \right\| = \left\| {({x_k} - x,0)} \right\| < \varepsilon $ or $\left\| {(0,{y_k} - y)} \right\| < \varepsilon $. If this is not true, a counter example is appreciated. Thanks!

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    @user251257 Thanks. I updated the question. – 2017-01-22
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    On a finite-dimensional real vector space, all norms are equivalent. Thus you can without loss of generality take $\lVert (x,y)\rVert_{V_1\times V_2} = \max \{ \lVert x\rVert_{V_1}, \lVert y\rVert_{V_2}\}$. – 2017-01-22

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