For each continuous $g:X\to $, $g(a_n)\to g(a)$, can we deduce $a_n\to a$?

Assume

  • $(X,\mathcal T)$ is a Tychonoff space.
  • $(a_n)$ is a sequence in $X$.
  • $a\in X$.
  • for each continuous function $g:X\to [0,1]$,
    $$g(a_n)\to g(a)$$

Is there an elementary proof for
$$a_n\to a$$
?

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