Question about Normed vector space.

Consider the following sequence of elements in the space $V$ of finite sequences: $$ u_1=(1,0,0,\ldots),\ \ u_2=(0,\frac12,0,\ldots),\ \ u_3=(0,0,\frac13,0,\ldots) $$ Then $$ \sum_{k=1}^nu_k=(1,\frac12,\frac13,\ldots,\frac1n,0,\ldots) $$ Now consider these two norms on $u=(a_1,a_2,\ldots)$: $$ \|u\|_1=\sum_{k=1}^\infty|a_k|,\ \ \ \|u\|_2=\left(\sum_{k=1}^\infty|a_k|^2\right)^{1/2} $$ Then, for the $u_n$ defined above, $$ \left\|\sum_{k=M}^nu_k\right\|_1=\sum_{k=M}^N\frac1k,\ \ \ \left\|\sum_{k=M}^Nu_k\right\|_2=\sum_{k=M}^N\frac1{k^2} $$ So, in $\|\cdot\|_1$, the tails of the series $\sum_{k=1}^\infty u_k$ are unbounded, which means that the series diverges; while in $\|\cdot\|_2$, the tails go to zero, so the series converges.