Why should we care about sequence in real analysis?

When first think about machine I can think of many useless machines which behave arbitrarily. For example, a random collection of levers, wheels, and pulleys connected by ropes doesn't do anything (i connect these pieces randomly). It seems that the number of useless machines is much more than that of useful machine.

So, does the concept of machine naturally arise in engineering ? Why is the concept of machine extensively while there are way more useless machines than the useful ones ?

There are various ways to interpret the question. Firstly, it is certainly the case that the amount of useless sequences far exceeds that of useful sequences. However, this will almost always be the case with any definition. The price you pay for putting lots of different interesting things under the same umbrella is that with them you also get lots of boring things under your umbrella. However, that is not a bug but rather feature of good definitions. The point is not to throw away the baby with the bath water. Simply allow all these useless exemplars co-exist with the ones that actually interest you. More importantly, just because we defined sequences and say things like "let $s$ be a sequence" rather than "let $s$ be an interesting sequence" does not mean we have any particular interest in all sequences. There will be toy examples that we'll look at just to clarify concepts but in practice we will always actually be looking at interesting sequences. Just don't try to visualise all sequences.

The above should answer the second half of your question. As for the first half, from an applications perspective sequences arise all the time. For instance, if you perform a measurement in physics the result you get is hardly ever precise. Too many factors affect your measurement so what you really always get is some rational number as the result of performing the measurement. Now, you can, and should if you are a good physicist, never accept the first measurement result as definitive. So, you improve your testing conditions and measure again. And again. And again. You do so a few dozen cases. What you obtain is a finite sequence. Of these values, which is the correct one? That is hard to tell, but we can idealise. If you had enough time you'd perform the measurement repeatedly forever, each time learning from the previous measurements and improving. This ideal situation leads to a genuine sequence. What is the true result of the measurement? Well, still hard to say but if the sequences converges, then there is some weight to the claim that the limit of the sequence is the true measurement result.

So, the above explains why the practicing user of analysis (e.g., physicists, engineers, chemists, etc.) are interested in sequences. They produce finite ones all the time and if only given enough time they would produce actual sequences. So it is a useful abstraction. (Incidentally, this touches upon a general principle in analysis whereby infinity is considered a very good approximation for large numbers.) But one may wonder whether sequences arise naturally also from an intrinsically mathematical perspective. The answer is yes, to some extent. It's true that the real numbers can be constructed by using sequences of rationals, but that is secondary (the reals can be constructed in many ways that do not mention sequences at all). Regardless of how the reals are constructed, or if they are constructed at all, the topology of the reals is strongly related to sequences. Convergence with respect to the natural topology of $\mathbb R$ is fully and faithfully captured by the convergence of sequences. This, however, is not true in all spaces that are directly of interest in analysis.

It's conceptually simpler to begin with the arbitrary sequence notion and then specify further as needed (e.g. monotone sequence, decreasing sequence, recursive sequence, integer sequence, positive sequence, monotone decreasing sequence, rational sequence, alternating sequence, convergent sequence, etc.) than it is to start with one of these as the "base notion" and then add or subtract properties as needed. For example, suppose "sequence" means what we now call a monotone decreasing sequence. What name would you propose that we give to what we now call an alternating sequence?