**Proof plan.** The forward direction ([continuity](/page/Continuity) implies sequential continuity) is immediate. For the reverse direction, suppose $T : V \to W$ is sequentially continuous. The strategy is to use the [Universal Property](/theorems/706) to reduce global continuity to continuity on each Fréchet piece $V_n$. On each piece, sequential continuity is equivalent to full continuity because $V_n$ is metrizable. The key link is the [Embedding Property](/theorems/707), which ensures that convergence of a sequence in $V_n$ implies convergence in $V$, allowing the sequential continuity hypothesis on $V$ to be inherited by each $V_n$. **Step 1: Forward direction.** If $T : (V, \tau_{\mathrm{ind}}) \to (W, \sigma)$ is continuous and $(x_k)_{k=1}^\infty$ is a sequence in $V$ with $x_k \to 0$ in $V$, then $T(x_k) \to T(0) = 0$ in $W$ by continuity. So continuity implies sequential continuity. **Step 2: Sequential continuity of $T$ on $V$ implies sequential continuity of $T|_{V_n}$.** [claim: Inheritance Of Sequential Continuity] If $T : V \to W$ is sequentially continuous, then for each $n \in \mathbb{N}$, the restriction $T \circ j_n : (V_n, \tau_n) \to (W, \sigma)$ is sequentially continuous. [/claim] [proof] Let $(x_k)_{k=1}^\infty$ be a sequence in $V_n$ with $x_k \to 0$ in $(V_n, \tau_n)$. By the [Embedding Property](/theorems/707), $j_n : (V_n, \tau_n) \to (V, \tau_{\mathrm{ind}})$ is a [topological](/page/Topology) embedding, so $j_n(x_k) \to j_n(0) = 0$ in $(V, \tau_{\mathrm{ind}})$. By the sequential continuity of $T$ on $V$, $T(j_n(x_k)) \to T(0) = 0$ in $(W, \sigma)$. Hence $T \circ j_n$ is sequentially continuous on $(V_n, \tau_n)$. [/proof] **Step 3: Sequential continuity on a [Fréchet space](/page/Fr%C3%A9chet%20Space) implies continuity.** [claim: Fréchet Sequential Continuity] Let $(E, \tau)$ be a Fréchet space, $(W, \sigma)$ a locally convex space, and $S : E \to W$ a sequentially continuous [linear map](/page/Linear%20Map). Then $S$ is continuous. [/claim] [proof] Since $(E, \tau)$ is metrizable, the topology is determined by [sequences](/page/Sequence): a [set](/page/Set) $A \subseteq E$ is closed if and only if it is sequentially closed. Let $C \subseteq W$ be $\sigma$-closed. The preimage $S^{-1}(C)$ is sequentially closed in $E$: if $(x_k)_{k=1}^\infty \subseteq S^{-1}(C)$ and $x_k \to x$ in $E$, then $S(x_k) \to S(x)$ in $W$ by sequential continuity, and $S(x_k) \in C$ for all $k$. Since $C$ is closed, $S(x) \in C$ and $x \in S^{-1}(C)$. In a metrizable space, sequentially [closed sets](/page/Closed%20Set) are closed, so $S^{-1}(C)$ is $\tau$-closed. Since the preimage of every closed set is closed, $S$ is continuous. [/proof] **Step 4: Conclusion via the universal property.** By Step 2, $T \circ j_n$ is sequentially continuous on the Fréchet space $(V_n, \tau_n)$ for each $n \in \mathbb{N}$. By Step 3, $T \circ j_n$ is continuous for each $n$. By the [Universal Property](/theorems/706), $T : (V, \tau_{\mathrm{ind}}) \to (W, \sigma)$ is continuous.