Lemma 3

Let E be a metrisable locally convex space and $(x_n)_{n\in\mathbf{N}}$ a sequence of elements of E converging to 0. There exist a sequence $(y_n)_{n\in\mathbf{N}}$ of elements of E converging to 0 and a sequence $(\lambda_n)_{n\in\mathbf{N}}$ of elements of the interval $[0,1]$ converging to 0 such that one has $x_n=\lambda_ny_n$ for every $n\in \mathbf{N}$.

Since E is metrisable, there exists a fundamental system $(V_m)_{m\in\mathbf{N}}$ of balanced neighbourhoods of 0 in E such that $V_0= E$ and $2^{m+1}V_{m+1}\subset V_m$ for every $m\geqslant 0$. Since $(x_n)$ converges to 0, there exists a strictly increasing sequence $(N_m)_{m\in\mathbf{N}}$ of integers such that $N_0= 0$ and such that, for every $m\geqslant 0$, one has $x_n\in V_m$ for $n\geqslant N_m$. For every integer $n\geqslant 0$, there exists a unique integer $m\geqslant 0$ such that $N_m\leqslant n <N_{m+1}$, and one then puts $\lambda_n= 2^{-m}$ and $y_n= 2^mx_n$. The sequence $(\lambda_n)$ thus defined converges to 0. Moreover, since $y_n\in V_m$ for $n\geqslant N_{m+1}$, the sequence $(y_n)$ converges to 0 in E. Finally, one has $x_n=\lambda_ny_n$ for every $n$.

Recall (EVT, II, p. 28) that if E is a vector space and L a convex balanced subset of E, one denotes by $E_L$ the vector subspace of E generated by L, endowed with the seminorm whose unit ball is L. When the set L contains no line, this seminorm is a norm.