Constructing an Increasing Sequence to Prove a Limit is Positive Infinity

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SUMMARY

The discussion centers on proving the existence of an increasing sequence {x_n} in the interval (a,b) such that f(x_n) > n, given that lim(x->b-) f(x) = +∞. Participants emphasize the importance of understanding the definition of the limit approaching positive infinity and suggest that the function f does not need to be continuous or strictly increasing. The key takeaway is that the limit condition guarantees the existence of such a sequence, despite the potential discontinuities of f.

PREREQUISITES
  • Understanding of limits in calculus, specifically lim(x->b-) f(x) = +∞
  • Familiarity with sequences and their properties in real analysis
  • Knowledge of increasing functions and their definitions
  • Basic concepts of continuity and discontinuity in functions
NEXT STEPS
  • Study the formal definition of limits approaching positive infinity in real analysis
  • Explore the properties of increasing sequences and their convergence
  • Investigate examples of functions that are not continuous but still satisfy limit conditions
  • Learn about the Bolzano-Weierstrass theorem and its implications for sequences in bounded intervals
USEFUL FOR

Mathematics students, particularly those studying real analysis, educators teaching calculus concepts, and anyone interested in the properties of limits and sequences in mathematical proofs.

Icebreaker
Suppose f:[a,b)->R is such that lim(x->b-)=+inf. Prove that there exists an increasing sequence {x_n} in (a,b) such that f(x_n)>n for all n.

I don't know where to start. It would be easy if I can prove that f is strictly increasing after some point. f might not be continuous so I can't simply look for all f(x) in the form x^2 or something... Too many possibilities. Any pointers will be helpful!
 
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What do you have for a definition of lim(x->b-)=+inf ? This will tell you where to start.
 

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