real analysis - Limit of $left(1-frac{1}{2^2}right)left(1-frac{1}{3^3}right)dots left(1-frac{1}{n^n}right)$ as $nto infty$

So I'm trying to solve the following limit:

$$\lim_{n \to \infty}\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^3}\right)\dots \left(1-\frac{1}{n^n}\right)$$

Now, I tried getting the squeeze theorem around this one, since it does feel like something for the squeeze theorem. The upper bound is obviously $1$, but since each term decreases the product, it may seem like this approaches zero?

Answer

$$\begin{align} \prod_{n=2}^\infty\left(1-\frac1{n^n}\right) &\ge\prod_{n=2}^\infty\left(1-\frac1{n^2}\right)\\ &=\lim_{m\to\infty}\prod_{n=2}^m\frac{n-1}n\frac{n+1}n\\ &=\lim_{m\to\infty}\prod_{n=2}^m\frac{n-1}n\prod_{n=2}^m\frac{n+1}n\\ &=\lim_{m\to\infty}\frac1m\frac{m+1}2\\[6pt] &=\frac12 \end{align}$$
So the product converges (that is, it is bounded away from $0$). However, numerically, the product is approximately $0.71915450096501024665446931$ and the Inverse Symbolic Calculator does not find anything for this number.