Banach space for Frobenius norm

Question

I want to prove that  $(mathcal{M}_{mtimes n}(mathbf{R}), ||.||_F)$ is a Banach Space
where
$A in mathcal{M}_{mtimes n}(mathbf{R})$, the Frobenius norm:
$$
||A||_F = sqrt{sum_{i = 1}^{m} sum_{j=1}^{n} a_{ij}^{2} } 
$$
I attempted to redefine the matrix $A$ into linear mapping $T: mathbf{R}^{n} to mathbf{R}^{m}$, where $T(x) = Ax$ for $x in mathbf{R}^{n}$.
I am stuck on showing a Cauchy Sequence $(T_n) subset mathbf{R}^{m}$ is converging to $T$ with respect to the norm $ ||. ||_F$ . How should I go from this approach or any suggestion differently?

Dong Le · Answer

So, should I define the Cauchy sequence of matrices in this way:
$$
A^{(n)} = 
begin{bmatrix}
    x_{11}^{(n)} & dots  & x_{1n}^{(n)} \
    vdots & vdots  \
    x_{m1}^{(n)} & dots  & x_{mn}^{(n)}
end{bmatrix}
$$
where each entries in these matrices are Cauchy Sequence. By Classical Convergence Cauchy criterion in $mathbf{R}$, it converges. So the Cauchy sequence of matrices converge to $A$ with respect to $||.||_F$.

Banach space for Frobenius norm

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