Lim sequence $\neq$ lim subsequence

Question

Let $\{x_n\}$ be a sequence and $\{y_k\}$ be a subsequence of $\{x_n\}$. Can you show me an example of $\{x_n\}$ such that $\displaystyle \lim_{k \to +\infty} y_k= +\infty$, but $\displaystyle \lim_{n \to +\infty} x_n\neq +\infty$ ?

Answer 1

This is possibly a bit of a notational problem. If a sequence $(x_n)$ of real numbers is convergent, it has a specific real number as its limit and we denote this limit with $\lim_{n\to\infty}x_n$. If a sequence is divergent, its limit doesn't even exist, so we should not assume that it is equal to anything. It is somewhat problematic if we can say that this nonexistent limit then is nonequal to anything. Since $(-1)^n$ does not converge, the statement $$\tag1\lim_{n\to\infty}(-1)^n=42$$ is clearly not a true statement. But does that make it a false statement and hence $$\tag2\lim_{n\to\infty}(-1)^n\ne42$$ a true statement? After all the expression on the left hand side is not even defined! Note that $$\tag3 \forall x\colon(\forall y\ne 666\colon x\ne y\to x=666)$$ is a theorem. If we accept that $(2)$ is true and is true if we replace $42$ with anything else (and we simply ignore the fact that it would also be true when replacing $42$ with $666$), then we find that in fact $$ \lim_{n\to\infty}(-1)^n=666.$$ This is of course not desireable.

One could circumvent this situation by making an extended definition of the limit operator, for example introduce a new object with the semantic of "undefined", define the limit of divergent sequences accordingly and so on. However, the introduction of a special symbol for undefined to solve the problem at hand would introduce new problems, for example in statements such as $$\tag4\lim_{n\to\infty}(a_n+b_n) = \lim_{n\to\infty}a_n+\lim_{n\to\infty}b_n. $$ It is much easier to be consistant if one accompanies such statements with "provided at least two of the limits exist" than to go on and try to introduce an arithmetic involving "undefined". To smaller extent, this is done with divergent sequences that tend to infinity. These do not have a limit, but nevertheless we write e.g. $$\lim_{n\to\infty}x_n=+\infty $$ either by "abuse of notation" or by introducing extended real numbers (and have some minor problems with $(4)$ as the arithmetic involving infinities is not always defined).

After these remarks, it is easy to find a sequence $(x_n)$ that is divergent and not tending to infinity and that has a subsequence $(y_n)$ that tends to infinity. For example, let $$ x_n = (1+(-1)^n)n\qquad y_n=x_{2n}=4n.$$ However, while this clearly (as remarked, by abuse of notation or using extended reals) gives us $$\lim_{n\to\infty} y_n=+\infty,$$ I'd be very hesitant to write $$\lim_{n\to\infty} x_n\ne\text{whatever (including $\infty)$}$$ for the reasons given above.