How to find and odd number nn s.t. nV⊂WnV\subset W?

Theorem 3.6 i) here (Pratulananda Das and Ekrem Savas: On II-convergence of nets in locally solid Riesz spaces, Filomat 27:1 (2013), 89â€“94, DOI:10.2298/FIL1301089D)

3.63.6 i) Iد„−limsخ±=x0â‡′Iد„−lima.sخ±=a.x0I_د„-\lim s_خ± = x_0 â‡’ I_د„-\lim a.s_خ± = a.x_0 for each a âˆˆ \mathbb Ra âˆˆ \mathbb R

proof: Let UU be a د„د„-neighborhood of zero. Choose V âˆˆ N_{sol}V âˆˆ N_{sol} such that V âٹ‚ UV âٹ‚ U. Since I-\lim s_خ± = x_0I-\lim s_خ± = x_0,
\{خ± âˆˆ D : s_خ± âˆ’ x_0 âˆˆ V\} âˆˆ F(I)\{خ± âˆˆ D : s_خ± âˆ’ x_0 âˆˆ V\} âˆˆ F(I).
Let |a| â‰¤ 1|a| â‰¤ 1. Since VV is balanced, s_خ± âˆ’ x_0 âˆˆ Vs_خ± âˆ’ x_0 âˆˆ V implies that a (s_خ± âˆ’ x_0) âˆˆ Va (s_خ± âˆ’ x_0) âˆˆ V.
Hence we have
\{خ± âˆˆ D : s_خ± âˆ’ x_0 âˆˆ V\} âٹ‚ \{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ V\} âٹ‚ \{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ U\}\{خ± âˆˆ D : s_خ± âˆ’ x_0 âˆˆ V\} âٹ‚ \{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ V\} âٹ‚ \{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ U\}

and so
\{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ U\} âˆˆ F (I).\{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ U\} âˆˆ F (I).

Now let |a| > 1|a| > 1 and as usual let [|a|][|a|] be the smallest integer greater than or equal to |a||a|. There exists a \color{blue}{W âˆˆ N_{sol} \text{
such that [|a|] W âٹ‚ V}}\color{blue}{W âˆˆ N_{sol} \text{
such that [|a|] W âٹ‚ V}}. Since I_د„-\lim s_خ± = x_0I_د„-\lim s_خ± = x_0,
A = \{خ± âˆˆ D : s_خ± âˆ’ x_0 âˆˆ W\} âˆˆ F (I)A = \{خ± âˆˆ D : s_خ± âˆ’ x_0 âˆˆ W\} âˆˆ F (I) .
Then we have
|a.x_0 âˆ’ a.s_خ±| = |a| |x_0 âˆ’ s_خ±| â‰¤ [|a|] |x_0 âˆ’ s_خ±| âˆˆ [|a|] W âٹ‚ V âٹ‚ U|a.x_0 âˆ’ a.s_خ±| = |a| |x_0 âˆ’ s_خ±| â‰¤ [|a|] |x_0 âˆ’ s_خ±| âˆˆ [|a|] W âٹ‚ V âٹ‚ U
for each خ± âˆˆ Aخ± âˆˆ A. Since the set VV is solid, we have a.s_خ± âˆ’ a.x_0 âˆˆ Va.s_خ± âˆ’ a.x_0 âˆˆ V and so a.s_خ± âˆ’ a.x_0 âˆˆ Ua.s_خ± âˆ’ a.x_0 âˆˆ U for each خ± âˆˆ Aخ± âˆˆ A. So we get
\{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ U\} âٹƒ A\{خ± âˆˆ D : a.s_خ± âˆ’ a.x_0 âˆˆ U\} âٹƒ A
and so it belongs to F(I)F(I). Hence I_د„-\lim a.s_خ± = a.x_0I_د„-\lim a.s_خ± = a.x_0 for every a âˆˆ \mathbb Ra âˆˆ \mathbb R

Now the \color{blue}{ coloured}\color{blue}{ coloured} part , how do we find that WW? From the definition of N_{sol}N_{sol} we can say we have V_1V_1 s.t V_1+V_1\subset WV_1+V_1\subset W and then V_2\in N_{sol}V_2\in N_{sol} s.t. V_2+V_2\subset V_1V_2+V_2\subset V_1 and thus V_2+V_2+V_2+V_2\subset WV_2+V_2+V_2+V_2\subset W and so on. going this way we can, at one stage, find that nV\subset WnV\subset W where nn is even number. What happens if |[a]||[a]| is odd?

Hope I could make my problem clear to you. pls help.thanks.

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You know that for each kk there exists VV such that
2^k V \subseteq W.2^k V \subseteq W.
(This can be done in the way you described.)

Now you can simply take kk large enough to have n\le 2^kn\le 2^k and you get
n V \subseteq 2^k V \subseteq W.n V \subseteq 2^k V \subseteq W.
(Just notice that if V\ni 0V\ni 0 and s\le ts\le t are positive integers, then sV\subseteq tVsV\subseteq tV. If we apply this for s=ns=n and t=2^kt=2^k, we get the above inclusion.)1

However, this is used to prove that \newcommand{\Ilim}{\operatorname{I-lim}}\Ilim s_\alpha =x_0\newcommand{\Ilim}{\operatorname{I-lim}}\Ilim s_\alpha =x_0 implies \Ilim as_\alpha=ax_0\Ilim as_\alpha=ax_0. And the proof given in the paper seems to be unnecessary complicated.

We know that f \colon x\mapsto axf \colon x\mapsto ax is a continuous function. (This is true for arbitrary topological vector space.)

So for any convergent net and any ideal we have
i.e.,

Similarly, to see that Theorem 3.6(ii) holds, we can just use continuity of addition as the map V\times V\to VV\times V\to V, and the claim immediately follows. (Basically, I am just saying that we do not need to reprove from scratch facts which are already known, or things which can be easily derived from some well-known facts.)

1Since (based on the comments) this part seemed unclear: If x\in sVx\in sV, this means that x=v_1+\dots v_sx=v_1+\dots v_s for some v_1,\dots,v_2\in Vv_1,\dots,v_2\in V. And we can now rewrite xx as
x=v_1+\dots v_s+\underset{\text{$(t-s)$-times}}{\underbrace{0+\dots+0}}x=v_1+\dots v_s+\underset{\text{$(t-s)$-times}}{\underbrace{0+\dots+0}}
to see that xx also belongs to tVtV.

Thank you. Actually after posting the question this occurred to me : VV and V\times \{0\}V\times \{0\} are homeomorphic and V\times \{0\} \subset V\times VV\times \{0\} \subset V\times V imply that V\subset V\times \{0\}V\subset V\times \{0\}.
– user118494
2 days ago

V\subset V\times\{0\}V\subset V\times\{0\} is certainly not true. The latter is a set of ordered pairs.
– Martin Sleziak
2 days ago

Sorry sorry. I meant V\subset V\times V.V\subset V\times V.
– user118494
2 days ago

That has still the same problem. (And it is not true in general.)
– Martin Sleziak
2 days ago

then how can we say nV\subseteq 2^kVnV\subseteq 2^kV for n\le 2^k\ ?n\le 2^k\ ?
– user118494
2 days ago