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Question Number 212741 by mathocean1 last updated on 22/Oct/24

a, b,c ∈ N^∗ Show that ab<c ⇒ a+b≤c

a,b,c∈N∗Showthatab<c⇒a+b⩽c

Answered by A5T last updated on 22/Oct/24

WLOG; let a≤b ⇒2a=a+a≤a+b≤b+b=2b≤ab<c when a≥2 ⇒a+b<c when a=1; ab=b<c⇒b+1≤c≡a+b≤c

WLOG;leta⩽b⇒2a=a+a⩽a+b⩽b+b=2b⩽ab<cwhena⩾2⇒a+b<cwhena=1;ab=b<c⇒b+1⩽c≡a+b⩽c

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