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Let’s see this sequence: ## s_n = \frac{3n+1}{7n-4}##. We need, not need I mean simply we... umm... we “would like to” find its limit.

I do this when I’m presented with dishes like that:

##s_n = \frac{3n+1}{7n-4}##, for very large ##n## that “1” and “4” won’t matter much, and that’s something very

*natural.*So, for large ##n##, we

*actually*have

$$

s_n = \frac{3n}{7n}= \frac{3}{7}$$

Thus, ##\lim ~ s_n = \frac{3}{7}##.

*##\epsilon-\delta## proof:*

Consider a sufficiently small ##\epsilon \gt 0##.

##\big| \frac{3n+1}{7n-4} -\frac{3}{7} \big| \lt \epsilon ##

##\big| \frac{19}{7(7n-4)} \big| \lt \epsilon##

As ##n \in \mathbf{N}##, we have ## 7n-4 \gt 0##. Therefore,

##\frac{19}{7(7n-4)} \lt \epsilon##

## \frac{19}{49 \epsilon} + \frac{4}{7} \lt n##

Take ##N = \frac{19}{49 \epsilon} + \frac{4}{7}##. By reversing the steps, we can establish the famous statement

**##n\gt N \implies \big| \frac{3n+1}{7n-4} - \frac{3}{7} \big| \lt \epsilon##**

For good reasons, the first workout is more

*rigorous*to me, I’m sure you will object “what did you really mean that 1 and 4 won’t matter much for large n?”, but that’s something we can

*perceive ,*I’m not advocating for empiricism but that ##\epsilon-\delta## proof doesn’t really tell me anything, it simply says “you can make ##s_n## as close to ##3/7## as you like”.

My argument that for very large n 1 and 4 won’t matter much, is something like the limitation of my eyes; if a helicopter were to take me slowly upwards: I would, surely, say shrubs don’t matter much and after a sufficient height I would really be unable to see those little shrubs and so, the only things that would be visible to me are trees.

If you could provide me the actual philosophy of Augustine Cauchy behind his “as close as you like”, the philosophy that he might have conveyed in those polytechnic college lectures, I would be grateful.