The Supplement for《Measure, Integration & Real Analysis》by Sheldon Axler is in the following page:
https://measure.axler.net/SupplementMIRA.pdf
Exercise 34 on p43 of the Supplement is the following:
Prove that if $F_1$ is a closed subset of $R^n$ and $F_2$ is a closed bounded subset of $R^n$, then $F_1 + F_2$ (which is defined to be $\{x + y : x ∈ F_1, y ∈ F_2\}$) is closed.
I try to devise $F_1$ and $F_2$ as the following:
$F_1 = \{x \,|\, x≥0, 0\, on\, other\, coordinates \}\\
F_2 = \{\frac{1}{x} \,|\, x∈ \Bbb N, 0\, on\, other\, coordinates \}$
It seems that:
$F_1 + F_2 = \{x \,|\, x>0, 0\, on\, other\, coordinates \}$
So it seems that:
A. $F_1$ is closed
B. $F_2$ is closed and bounded
C. $F_1 + F_2$ is not closed
What's wrong with the ABC above?