Null Sequence in Metrizable Topological Vector Space Dominates some Sequence of Scalars Tending to Infinity

Theorem

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \tau}$ be a metrizable topological vector space over $\GF$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence with $x_n \to {\mathbf 0}_X$.

Then there exists a sequence of positive real numbers $\sequence {\gamma_n}_{n \mathop \in \N}$ such that:

$\gamma_n \to \infty$

and:

$\gamma_n x_n \to {\mathbf 0}_X$

Then:

$\map d {x_n, { {\mathbf 0}_X} } \to 0$

Pick $n_1 \in \N$ such that:

$\map d {x_n, { {\mathbf 0}_X} } < 1$ for $n > n_1$.

Inductively, for $k \ge 2$, pick $n_k > n_{k - 1}$ such that:

$\map d {x_n, { {\mathbf 0}_X} } < k^{-2}$ for $n > n_k$.

For $n < n_2$, set $\gamma_n = 1$.

For $k \ge 2$ and $n_k \le n < n_{k + 1}$, set $\gamma_n = k$.

For each $n$, let $\map k n$ be the unique $k \in \N$ such that $n_k \le n < n_{k + 1}$.

Since $n_k < n_{k + 1}$, we have $\map k n \to \infty$ as $n \to \infty$.

Now, for $n \ge n_2$:

$\ds \map d {\gamma_n x_n, { {\mathbf 0}_X} }$

$=$

$\ds \map d {\map k n x_n, { {\mathbf 0}_X} }$

$\ds $

$\le$

$\ds \map k n \map d {x_n, { {\mathbf 0}_X} }$

$\ds $

$<$

$\ds \frac 1 {\map k n}$

since $\map d {x_n, { {\mathbf 0}_X} } < k^{-2}$ for $n > n_k$

Taking $n \to \infty$, we obtain:

$\map d {\gamma_n x_n, { {\mathbf 0}_X} } \to 0$

and hence:

$\gamma_n x_n \to {\mathbf 0}_X$

$\blacksquare$