Prove that (1 + 1/n)^n ﹤ 3 (ILIEKMATHPHYSICS)

In this video, we prove that for all positive integers n, (1 + 1/n)^n ﹤ 3. This is a useful component if you want to prove that the sequence (1 + 1/1)^1, (1 +  1/2)^2, (1 + 1/3)^3, ..., (1 + 1/n)^n, ... converges. In fact, this sequence converges to the value e = 2.718281828459.... This proof shows that this sequence is bounded above (by 3), but we did not show that the sequence is monotonically increasing. In fact, you can use a very similar argument (using Bernoulli's Inequality) to show that this sequence is monotonically increasing. By the Monotone Convergence Theorem, the sequence converges, and we may call the value it converges to by "e".

Notice that the method to prove (1 + 1/n)^n ﹤ 3 in this video can be extended to a stronger claim -- in the sense that you can also prove (1 + 1/n)^n ﹤ 2.75, or even (1 + 1/n)^n ﹤ 2.72.

The fact in this video is important in real analysis. My main real analysis playlist references the book "Intro to Real Analysis" by Bartle and Sherbert: Real Analysis (Bartle) [ILIEKMATHPHYS.... The book's approach to the limit of (1 + 1/n)^n is given in Section 3.3, and it is completely different from the way it is done here. There are many approaches out there regarding (1 + 1/n)^n.

Thanks and enjoy the video!