Convergence in Direct Product Norm

Theorem
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces.

Let $X \times Y$ be a direct product of vector spaces $X$ and $Y$ together with induced component-wise operations.

Let $\norm {\, \cdot \,}_{X \times Y}$ be the direct product norm.

Let $\sequence {\tuple {x_n, y_n} }_{n \mathop \in \N}$ be a sequence in $X \times Y$.

Then:


 * $\tuple {x_n, y_n}$ converges to $\tuple {x, y}$ in $X \times Y$




 * $x_n$ converges to $x$ in $X$

and:


 * $y_n$ converges to $y$ in $Y$.

Neccessary Condition
Suppose that:


 * $\tuple {x_n, y_n}$ converges to $\tuple {x, y}$ in $X \times Y$

Then, for each $\epsilon > 0$, there exists $N \in \N$ such that:


 * $\ds \norm {\tuple {x_n, y_n} - \tuple {x, y} }_{X \times Y} < \epsilon$

for all $n \ge N$.

This is equivalent to:


 * $\ds \norm {\tuple {x_n - x, y_n - y} }_{X \times Y} < \epsilon$

by the definition of induced component-wise operations.

From the definition of the direct product norm, this is equivalent to:


 * $\ds \map \max {\norm {x_n - x}_X, \norm {y_n - y}_Y} < \epsilon$

This is equivalent to:


 * $\norm {x_n - x}_X < \epsilon$

and:


 * $\norm {y_n - y}_Y < \epsilon$

for $n \ge N$.

That is:


 * $\sequence {x_n}_{n \mathop \in \N}$ converges to $x$ and $\sequence {y_n}_{n \mathop \in \N}$ converges to $y$.

Sufficient Condition
Suppose that:


 * $x_n$ converges to $x$ in $X$

and:


 * $y_n$ converges to $y$ in $Y$.

Let $\epsilon > 0$.

Then we can pick $N_1 \in \N$ such that:


 * $\norm {x_n - x}_X < \epsilon$

for $n \ge N_1$.

Similarly, we can pick $N_2 \in \N$ such that:


 * $\norm {y_n - y}_Y < \epsilon$

Let:


 * $N = \max \set {N_1, N_2}$

Then, for $n \ge N$, we have:


 * $\norm {x_n - x}_X < \epsilon$

and:


 * $\norm {y_n - y}_Y < \epsilon$

so that:


 * $\map \max {\norm {x_n - x}_X, \norm {y_n - y}_Y} < \epsilon$

From the definition of the direct product norm, we have:


 * $\norm {\tuple {x_n - x, y_n - y} }_{X \times Y} < \epsilon$

From the definition of induced component-wise operations, we have:


 * $\norm {\tuple {x_n, y_n} - \tuple {x, y} }_{X \times Y} < \epsilon$

for all $n \ge N$.

So:


 * $\tuple {x_n, y_n}$ converges to $\tuple {x, y}$ in $X \times Y$