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    Inverse composite proof (wording of the proof)

    Homework Statement Let f:A->B and g:B->C be invertible mappings. Show (g o f)^-1 = f^-1 o g^-1. Homework Equations A mapping is invertible iff it is bijective The Attempt at a Solution I understand why these are equivalent statements; however, I can't figure out the wording of the...
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    Proving the pigeonhole directly. I'm stuck.

    Homework Statement Prove the pigeonhole principle directly. so basically |Nk-{x}|=|Nk-1| if k>1 is an integer and x belongs to the set of naturals. Homework Equations The Attempt at a Solution I have no idea even where to begin.
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    A quick question on Summations.

    I can never remember how to expand a summation in to form: \sumnk=1(22). Thats just a recent example. I can't remember the expansion form any sort of summation really except when it has a defined upper bound.
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    A quick question about Butane as a solvent?

    If say cold butane bound to something ( butane as a solvent ) is poured into warm water, then what happens. I don't believe anything will happen except the butane will boil off and what is in the solution will fall off to the bottom till the water is cooled completely but I'm just checkin with...
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    Kinda Urgent. Proof Wriiting. Squences

    Homework Statement Prove that if X[n]->L, X[n] doesn't equal 0 for all n, L doesn't equal 0 and Y[n]->M, then (Y[n]/X[n])->(M/L). Homework Equations They only give you the squeeze theroem and that if X[n] converges then it's limit is unique. O and the definition. A sequence of reals has...