Let $(X,||.||)$ a banach space. Let $Y,Z$ closed subspaces of $X$ such that $X=Y\oplus Z$ (algebraic direct sum). For $x=y+z$ with $y\in Y$ and $z\in Z$ we define:
$$||x||_0=||y||+||z||$$
Prove that $(X,||.||_0)$ is banach space.
My attempt:
Let $\{x_n\}\subset X$ with norm $||.||_0$ a cauchy sequence.
As $Y$ and $Z$ are closed subset of $X$ and $(X,||.||)$ is a banach space, then $Y,Z$ are banach.
Let $\epsilon >0$, $\exists N\in\mathbb{N}$ such that if $n,m>N$ then $$||x_n-x_m||_0<\epsilon$$
Moreover,
$$||x_n-x_m||_0=||y_n-y_m||+||z_n-z_m||<\epsilon$$
This implies:
$$||y_n-y_m||<\epsilon\tag1$$
$$||z_n-z_m||<\epsilon\tag2$$
Then, $\{y_n\}$,$\{z_n\}$ are cauchy sequences in $Y,Z$.
As $Y,Z$ are banach spaces then $y_n\rightarrow y$ and $z_n\rightarrow z$ where$y\in Y$ and $z\in Z$
Here i'm stuck. can someone help me?