Infimum of decreasing sets is increasing?

Question

I have read that if we have a decreasing sequence of sets i.e. $\{A_{i}\}_{1}^{\infty}$, and $A_{1} \supset A_{2} \supset \dots$

If we take the infimum over this decreasing sequence of sets it is increasing? I don't understand this, to me it seems that the infimum would be decreasing...

Could someone please explain.

Thanks.

Answer 1

Hint: Try the sequence $[n,\infty) \subseteq \mathbb{R}$ for $n \in \mathbb{N}$.

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Also note that this is the case if by "take the infimum over this decreasing sequence of sets" you mean the sequence $\inf\cap_{i = 1}^n A_n$. If however you mean $\inf\cup_{i = 1}^n A_n$ then the sequence stays constant. Have a look at the example above.