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I have a question on limits:

We write $\lim_{n\to\infty}\frac{1}{n}=0$, even though the sequence $(x_n)$ defined by $x_n=\frac{1}{n}$ never really takes the value $0$. Right, so this is how we learned it. However, when working with sets, it seems to me we don't work with the same rules, because

$\lim_{n\to\infty}\{X\leq m-\frac{1}{n}\}\neq\{X\leq m\}$, but $\{X

even though $\lim_{n\to\infty}(m-\frac{1}{n})=m$.

So why do we make this apparent distinction?

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    It is just because the definition of “limit of a sequence of sets” is distinct from “limit of sequence of real numbers.” We say $\{X \leq m - 1/n\}\nearrow \{X < m\}$ because the latter is the set of points in infinitely many of the sets $\{X \leq m - 1/n\}_{n=1}^{\infty}$, and is also the set of points in all but a finite number of those sets. Sets allow fine-grain distinctions: The two sets “points less than $m$” and “points less than or equal to $m$" both exist. However, “the largest number less than or equal to $m$” exists, while “the largest number less than $m$” does not exist. – 2017-01-22
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    Thank you especially for pointing out why it makes sense to use different definitions. – 2017-01-22

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With a sequence of reals, you are rarely able to replace part of the limit with its own limit; for example, $\lim_{x \to 0}\frac{x}{x}$ can't be computed by taking $\lim_{x \to 0}\frac{x}{x} = \lim_{x \to 0}\frac{0}{x} = \lim_{x \to 0}0 = 0$, even though $\lim_{x \to 0}x = 0$. So why would you expect that you could do that with a limit of sets?

In any case, the two aren't really related, except that they both use the word "limit" - it's like asking why we eat "ears" of corn but not ears of people. The limit of a sequence of reals is defined using $\epsilon$s and absolute values and such, while the limit of a set is defined using set membership - carrying the analogy, one's a plant while the other's a person.