Let ( a n ) n ? N and ( b n ) n ? N be two Cauchy sequences. Prove that the sequ

Question

Let (an)n?N and (bn)n?N be two Cauchy sequences. Prove that the sequence (cn)n?N defined by cn = an + bn for all n, is also Cauchy.
Let ? > 0. Define ? ? = ?/2. Since (an)n?N and (bn)n?N are Cauchy there exist N1 and N2 such that:

?n>N1 and ?m>N1 we have |an ?am|<? ?,
and also
?n>N2 and?m>N2 wehave|bn?bm|<? ?.DefineN=max{N1,N2}. Let n>N and let m>N. We have:

|cn ?cm|=|(an ?am)+(bn ?bm)|?|(an ?am)|+|(bn ?bm)|<? ?+? ?=?.

" DefineN=max{N1,N2}"

I just don't know what this part means.

Explanation / Answer

Just look at N1 and N2. Then either N1 > N2 or N1<N2 or N1=N2.

So choose the maximum one among N1 and N2, call it as N,

Then any integer which is greater than N will also be greater than N1 and N2.

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.