In considering the theorem cited here uniform continuity and equivalent sequences , which states that where $f:X \rightarrow \mathbb{R}$ is a function, the following two conditions are equivalent:
(a) $f$ is uniformly continuous
(b) If $(x_{n})_{n=0}^{\infty}$ and $(y_{n})_{n=0}^{\infty}$ are equivalent sequences of real numbers, then $(f(x_{n}))_{n=0}^{\infty}$ and $(f(y_{n}))_{n=0}^{\infty}$ are also equivalent sequences of real numbers.
I was curious as to whether a direct proof could be offered for $(b) \rightarrow (a)$, and if not, how one would go about proving this. (I don’t have a sufficiently thorough background in intuitionistic logic to offer the latter kind of proof so wanted to solicit answers from whose who do). Any insight would be greatly appreciated!