Thread: convergence: proofs and counterexamples

For each of the following, prove or give a counterexample:

A) If (Sn) and (Tn) are divergent sequences, then (Sn + Tn) diverges
B) If (Sn) and (Tn) are divergent sequences, then (SnTn) diverges
C) If (Sn) and (Sn + Tn) are convergent sequences, then (Tn) converges
D) If (Sn) and (SnTn) are convergent sequences, then (Tn) converges

Originally Posted by luckyc1423

For each of the following, prove or give a counterexample:

A) If (Sn) and (Tn) are divergent sequences, then (Sn + Tn) diverges

False consider,
s_n=1,0,1,0,1,0,...
t_n=-1,0,-1,0,-1,0,...

B) If (Sn) and (Tn) are divergent sequences, then (SnTn) diverges

False consider,
s_n=t_n=(-1)^n/n

C) If (Sn) and (Sn + Tn) are convergent sequences, then (Tn) converges

If {s_n} converges and {s_n+t_n} convergesn then by theorems we have that,
{s_n+t_n}-{s_n}={t_n} converges.

D) If (Sn) and (SnTn) are convergent sequences, then (Tn) converges

No.

If s_n does not converges to zero then this is true.

But sometimes no.

Let s_n=0 and t_n=n.