A non-convergent sequence in R for which every convergent subsequences converges to 1

Question

I was told such a sequence could exist but didn't believe it. Is an example of the title statement possible?

Answer 1

The sequence $1,2,3,\dots$ is one such sequence.

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Or, if you want a sequence which actually has a convergent subsequence, take

$$1,\frac{3}{2}, 2,\frac43,3,\frac54,4,\frac65,\dots$$

Answer 2

If a trivial example such as $1,2,3,...$ does not satisfy you because it has no convergent sequences, then consider $1,2,1,3,1,4,1,5,1,...$ and note that any sequence of integers converges if and only if it is eventually constant(that is, there is some large $N$ such that the sequence $a_{n+N}$ is constant).

Therefore any convergent sequence must converge to $1$, since every other integer only comes once(or never) in the sequence.

But the big sequence itself doesn't converge for the same reason as above : let alone being constant, no two consecutive integers are even the same in the sequence. Hence, we see that the sequence itself does not converge.