We say that a sequence $\left( A ^ { ( k ) } \right) _ { k \in \mathbb { N } }$ of matrices in $M _ { n } ( \mathbb { C } )$ converges to a matrix $B \in M _ { n } ( \mathbb { C } )$ when $$\forall i \in \llbracket 1 , n \rrbracket , \forall j \in \llbracket 1 , n \rrbracket , \lim _ { k \rightarrow + \infty } \left( a _ { i , j } \right) ^ { ( k ) } = b _ { i , j }$$ Show that the sequence $( A ^ { ( k ) } )$ converges to $B$ if and only if $\lim _ { k \rightarrow + \infty } \left\| A ^ { ( k ) } - B \right\| = 0$.
We say that a sequence $\left( A ^ { ( k ) } \right) _ { k \in \mathbb { N } }$ of matrices in $M _ { n } ( \mathbb { C } )$ converges to a matrix $B \in M _ { n } ( \mathbb { C } )$ when
$$\forall i \in \llbracket 1 , n \rrbracket , \forall j \in \llbracket 1 , n \rrbracket , \lim _ { k \rightarrow + \infty } \left( a _ { i , j } \right) ^ { ( k ) } = b _ { i , j }$$
Show that the sequence $( A ^ { ( k ) } )$ converges to $B$ if and only if $\lim _ { k \rightarrow + \infty } \left\| A ^ { ( k ) } - B \right\| = 0$.