Sequence on Product Space Converges to Point iff Projections Converge to Projections of Point

Theorem
Let $N \in \N$.

For all $k \in \set {1, \ldots, N}$, let $T_k = \struct {X_k, \tau_k}$ be topological spaces.

Let $\displaystyle X = \prod_{k \mathop = 1}^N X_k$ be the cartesian product of $X_1, \ldots, X_N$.

Let $\tau$ be the product topology on $X$.

Denote by $\pr_k : X \to X_k$ the projection from $X$ onto $X_k$.

Let $\sequence {x_n}$ be a sequence on $X$ and let $x \in X$.

Then $\sequence {x_n}$ converges to $x$ :
 * for all $k \in \set {1, \ldots, N}$ the sequence $\sequence {\map {\pr_k} {x_n} }$ converges to $\map {\pr_k} x$.

Necessary Condition
Let $x_n \to x$.

Let $k \in \set {1, \ldots, N}$.

From Projection from Product Topology is Continuous it follows that $\pr_k$ is continuous.

By Continuous Mapping is Sequentially Continuous, $\pr_k$ is also sequentially continuous.

Hence $\map {\pr_k} {x_n} \to \map {\pr_k} x$.

Sufficient Condition
Let $\map {\pr_k} {x_n} \to \map {\pr_k} x$ for all $k \in \set {1, \ldots, N}$.

Let $U \in \tau$ be an open neighborhood of $x$.

By definition of the product topology and Synthetic Basis and Analytic Basis are Compatible it follows that:


 * $\BB := \set {U_1 \times U_2 \times \cdots \times U_N : \forall k \in \set {1, \ldots, N} : U_k \in \tau_k}$

is an analytic basis for $\tau$.

Hence there exists an index set $I$ such that:


 * $\displaystyle U = \bigcup_{i \mathop \in I} \paren {U_{1, i} \times \cdots \times U_{N, i} }$

where $U_{k, i} \in \tau_k$ for all $i \in I, k \in \set {1, \ldots, N}$.

As $x \in U$ it follows that there exists $i_0 \in I$ such that:


 * $\displaystyle x \in U_{1, i_0} \times \cdots \times U_{N, i_0}$

By our hypothesis $\map {\pr_k} {x_n} \to \map {\pr_k} x$ it follows that:


 * $\forall k \in \set{1, \ldots, N}: \exists M_k \in \N : \forall n \ge M_k: \map {\pr_k} {x_n} \in U_{k, i_0}$

Thus for all $n \ge M := \max \set {M_1, \dotsc, M_N}$ it holds that:


 * $ x_n = \tuple {\map {\pr_1} {x_n}, \dotsc, \map {\pr_N} {x_n} } \in U_{1,i_0} \times \cdots \times U_{N, i_0} \subset U$

Hence the result.

Also see

 * Filter on Product Space Converges to Point iff Projections Converge to Projections of Point