So far, I think that $\inf(A) = 7$ and that there is no supremum. I'm just not too sure how to prove this.
2026-04-06 16:57:06.1775494626
Prove that the upper bound, supremum, infimum and lower bound of the set $A=\{7+\frac{9}{5x}~|~ x ≥2\}$ exist or do not exist.
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You must realise that any monotonically decreasing sequence has a lower bound, from the Monotone Convergence Theorem. Hence, a lower bound will surely exist. Now, you can show by induction that 7 will be a lower bound, and the greatest one at that (Take any $M \gt 7$, and see what happens to your induction hypothesis)