Let $ \left ( X_{1},\left \| \right \|_{1} \right ) $,$ \left ( X_{2},\left \| \right \|_{2} \right ) $ two Banach spaces in the vector space X.
How to prove that $\left \| x \right \|= \inf \left \{\left \| x_{1} \right \|_{1}+\left \| x_{2} \right \|_{2}:x=x_{1}+x_{2} \ , \ x_{1}\in X_{1} \ \ x_{2}\in X_{2} \right \} $ defines a norm in $ X_{1}+X_{2} $
My attempt : $ \left \| x \right \|=0 \Leftrightarrow \exists \left ( x_{n} \right )_{n\in \mathbb{N}}\subset X_{1} \ , \exists \left ( y_{n} \right )_{n\in \mathbb{N}}\subset X_{2}: x=x_{n}+y_{n} \ and \ \lim_{n\rightarrow \infty}\left \| x_{n} \right \|_{1}=0 \ $ and $ \lim_{n\rightarrow \infty}\left \| y_{n} \right \|_{2}=0 $ I can't prove that $x=0$
Picking up from where you left off, if $\|x_n\|_1\to 0$, then $$ \|x_n-x_m\|_1\le \|x_n\|_1+\|x_m\|_1 \to 0$$ when $n,m\to\infty$. Thus, $(x_n)_{n\in\mathbb{N}}$ is Cauchy in $X_1$, and thus converges to a point, say $z$. The functions $v\mapsto \|v\|_1$ is continuous, so $\|z\|_1=\lim_n \|x_n\|=0$. Thus $z=0$.
By the same reasoning, $y_n\to 0$ in $X_2$.
Finally,
$$ x=\lim_{n\to\infty} x_n+y_n=0 $$ as desired.