Following are the four examples of sequences (along with their properties) which can be helpful to gain a better understanding of theorems about sequences (real analysis):

- : unbounded, strictly increasing, diverging
- : bounded, strictly decreasing, converging
- : bounded, strictly increasing, converging
- : bounded, not converging (oscillating)

I was really amazed to found that is a strictly increasing sequence, and in general, the function defined for all positive real numbers is an increasing function bounded by 1:

The graph of x/(1+x) for x>0, plotted using SageMath 7.5.1

Also, just a passing remark, since for all , and as seen in prime number theorem we get an unbounded increasing function for

The plot of x/log(x) for x>2. The dashed line is y=x for the comparison of growth rate. Plotted using SageMath 7.5.1

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