- #1
cragar
- 2,552
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Homework Statement
Assume [itex] x_n [/itex] and [itex] y_n [/itex] are Cauchy sequences.
Give a direct argument that [itex] x_n+y_n [/itex] is Cauchy.
That does not use the Cauchy criterion or the algebraic limit theorem.
A sequence is Cauchy if for every [itex] \epsilon>0 [/itex] there exists an
[itex] N\in \mathbb{N} [/itex] such that whenever [itex] m,n\geq N [/itex]
it follows that [itex] |a_n-a_m|< \epsilon [/itex]
The Attempt at a Solution
Lets call [itex] x_n+y_n=c_n [/itex]
now we want to show that [itex] |c_m-c_n|< \epsilon [/itex]
Let's assume for the sake of contradiction that
[itex] c_m-c_n> \epsilon [/itex]
so we would have
[itex]|x_m+y_m-x_n-y_n|> \epsilon [/itex]
[itex] x_m> \epsilon+y_n-y_m [/itex]
since [itex] y_n>y_m [/itex]
and we know that [itex] x_m< \epsilon [/itex]
so this is a contradiction and the original statement must be true.