Let $A$ be a $C^*$ Algebra. Let $J$ be a closed ideal in A . Let $B$ be $C^*$-sub-algebra of $A$. Prove $B+J$ is complete space (i.e. every cauchy sequence in $B+J$ converges to an element of $B+J$).
Denote by $x_n + y_n$ a Cauchy sequence in $B+J$ ($x_n \in B$, $y_n \in J$). We need to show $x_n$ is Cauchy sequence in $B$ and $y_n$ is Cauchy sequence in $J$. It seems we need to use the fact $J$ is ideal, But how ?
Any hints ?