We consider a sequence $\left( A _ { n } \right) _ { n \geqslant 0 }$ $$A _ { n } = \left( \begin{array} { l l } a _ { n } & b _ { n } \\ b _ { n } & d _ { n } \end{array} \right)$$ of symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We say that $S \leqslant S'$ if and only if $S' - S \geqslant 0$. We assume that the sequence $\left( A _ { n } \right) _ { n \geqslant 0 }$ is increasing and bounded for this order relation. II.H.1) Prove that for every vector $X$, the sequence $\left( { } ^ { t } X A _ { n } X \right) _ { n \geqslant 0 }$ is increasing and bounded. II.H.2) Prove that the sequences $\left( a _ { n } \right) _ { n \geqslant 0 }$ and $\left( d _ { n } \right) _ { n \geqslant 0 }$ are increasing and bounded. II.H.3) By considering the vector $X = ( 1,1 )$, prove that the sequence of matrices $\left( A _ { n } \right) _ { n \geqslant 0 }$ is convergent in $\mathcal { M } _ { 2 } ( \mathbb { R } )$, that is, the sequences $\left( a _ { n } \right) _ { n \geqslant 0 }$, $\left( b _ { n } \right) _ { n \geqslant 0 }$ and $\left( d _ { n } \right) _ { n \geqslant 0 }$ are convergent in $\mathbb { R }$.
We consider a sequence $\left( A _ { n } \right) _ { n \geqslant 0 }$
$$A _ { n } = \left( \begin{array} { l l } a _ { n } & b _ { n } \\ b _ { n } & d _ { n } \end{array} \right)$$
of symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We say that $S \leqslant S'$ if and only if $S' - S \geqslant 0$. We assume that the sequence $\left( A _ { n } \right) _ { n \geqslant 0 }$ is increasing and bounded for this order relation.
II.H.1) Prove that for every vector $X$, the sequence $\left( { } ^ { t } X A _ { n } X \right) _ { n \geqslant 0 }$ is increasing and bounded.
II.H.2) Prove that the sequences $\left( a _ { n } \right) _ { n \geqslant 0 }$ and $\left( d _ { n } \right) _ { n \geqslant 0 }$ are increasing and bounded.
II.H.3) By considering the vector $X = ( 1,1 )$, prove that the sequence of matrices $\left( A _ { n } \right) _ { n \geqslant 0 }$ is convergent in $\mathcal { M } _ { 2 } ( \mathbb { R } )$, that is, the sequences $\left( a _ { n } \right) _ { n \geqslant 0 }$, $\left( b _ { n } \right) _ { n \geqslant 0 }$ and $\left( d _ { n } \right) _ { n \geqslant 0 }$ are convergent in $\mathbb { R }$.