# Thread: counter examples in real analysis

1. ## counter examples in real analysis

For each of the following, prove or give a counter example:
a) if (Sn) converges to S, then (|Sn|) converges to |S|
b) if (|Sn|) is convergent, then (Sn) is convergent
c) lim Sn = o iff lim |Sn| = 0

2. Originally Posted by learn18
For each of the following, prove or give a counter example:
a) if (Sn) converges to S, then (|Sn|) converges to |S|
b) if (|Sn|) is convergent, then (Sn) is convergent
c) lim Sn = o iff lim |Sn| = 0
b) counter example Sn=(-1)^n, then |Sn| converges to 1, and Sn does not converge

RonL

3. Originally Posted by learn18
For each of the following, prove or give a counter example:
a) if (Sn) converges to S, then (|Sn|) converges to |S|
b) if (|Sn|) is convergent, then (Sn) is convergent
c) lim Sn = o iff lim |Sn| = 0
c) Suppose lim(n->inf) Sn = 0, this means that for all epsilon in R there exists
a natural number N such that:

|Sn - 0| < epsilon, for all n>N.

but this last is the same thing as:

|Sn| < epsilon, for all n>N

which is again the same as:

||Sn|-0| < epsilon, for all n>N

so lim(n->inf) Sn = 0 implies that lim(n->inf) |Sn| = 0.

For the other half of the proof, suppose lim(n->inf) |Sn| = 0, this means that
for all epsilon in R there exists a natural number N such that:

||Sn| - 0| < epsilon, for all n>N.

but this last is the same thing as:

|Sn| < epsilon, for all n>N

which is again the same as:

|Sn-0| < epsilon, for all n>N

so lim(n->inf) |Sn| = 0 implies that lim(n->inf) Sn = 0.

Together these proove: lim Sn = 0 iff lim |Sn| = 0

RonL

4. Originally Posted by learn18
For each of the following, prove or give a counter example:
a) if (Sn) converges to S, then (|Sn|) converges to |S|
if (s_n) converges to s then,
|s_n - s| < e for n>N

Note that,
||x|-|y||<=|x-y|

Thus,
||s_n|-|s||<=|s_n-s|<e for n>N

Thus,
lim |s_n| = |s|