Prove by induction

See attachment Prove by induction that for any and

Limits

see attachment a. Use the definition of limits to show that b. Compute the following limits (justify your answers). 1. ( ) 2. c. Let ( be a sequence. Show that ( is convergent if and only if the sequence ( and ( are convergent to the same limit.

Sequences

1. Define the following sequence . a. Show that and are monotone sequences. b. Show that and converge to the same limit. c. Find . d. If for some . Is ( ) still convergent. (see attachment)

Cauchy Condensation Criterion and Convergent Series

Please help me with the attached questions. Thanks.

Cauchy Sequences

See the attached document please. Thanks.