If you show that convergence of nets in a topological vector space $V$ with topology $\tau$ is equivalent to convergence of nets in a topological vector space $V$ With topology $\sigma$, does it necessarily follow that $\tau = \sigma$?
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You may find this easier to prove by showing that $\sigma$ and $\tau$ have the same closed sets, since closed sets are easier to describe in terms of nets than open sets are.