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Is the set of all strictly increasing sequences of positive integers whose consecutive terms grow not more than polynomial rate countable?

We know than set of all strictly increasing sequences of positive integers are uncountable but here we have restriction $|X_{n+1}-X_n|=O(x^m)$ for some $m\in N.$

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    Isn't the set of everything integer-like countable? – 2017-01-21
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    i did not get you – 2017-01-21
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    `We know than set of all strictly increasing sequences of positive integers are uncountable`. I thought that would be countable... – 2017-01-21
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    I have also same intuition but i want to prove it – 2017-01-21
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    @SimplyBeautifulArt: No, the set of infinite binary strings is uncountable, being the same (almost) as the set of reals between $0$ and $1$ – 2017-01-21
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    @RossMillikan Ah, that makes more sense. I'm sorry, you are right :D – 2017-01-21

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Consider the set of all strictly increasing sequences of positive integers with first term $1$ such that the gaps between successive terms is either $1$ or $2$. It's clear that the set of such sequences is uncountable (one-to-one correspondence to the set of infinite binary strings). But the $n$-th term of any such sequence is at most $2n + 1$.