Getting acquainted with compact sets

Task number: 3171

Determine whether the following subsets in \(\mathbb R^n\) are compact (justify your answer):

  • Variant 1

    \([0, 1]\times[0, 1]\times[0, 1]\)

  • Variant 2

    \(\{(x, y, z) \in \mathbb R^3: |x + y + z| \leq 1\}\)

  • Variant 3

    \(\{(x, y, z) \in \mathbb R^3: |x + y| + z \leq 1\}\)

  • Variant 4

    \(\{(x, y) \in \mathbb R^2: x^4 + y^4 < 10\}\)

  • Variant 5

    \(\{\frac1n: n \in \mathbb N\} \cup \{0\}\)

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
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