The question given in my book goes exactly as follows:
Let $X$ consist of all sequences $x=\{x_n\}$ of all real sequences, such that all but finitely many terms $x_n$ are zero. Prove that $d(x,y)=\bigg[\displaystyle\sum_{n=1}^\infty(x_n-y_n)^2\bigg]^{1/2}$ is a metric on $X$.
Approach:
Claim: $X \subset l_1 \subset l_2$
Proof: By hypothesis, $\forall \ \mathbf{x}=\{x_n\} \in X$, $\exists \ k\in \mathbb{N}$ such that $x_n=0, \quad \forall \quad n \geq k$. [only finitely many terms are $ \neq 0$]
So, $\sum_{n=1}^\infty |x_n|=\sum_{n=1}^kx_n<\infty$. Which implies $\mathbf{x}\in l_1$.
The rest follows from the Minkowski's inequality, i.e. $(\sum_{n=1}^\infty |a_n+b_n|^2)^{1/2} \leq (\sum_{n=1}^\infty|a_n|^2)^{1/2}+(\sum_{n=1}^\infty |b_n|^2)^{1/2}$
Is this the correct approach? Please verify.