Proving Sequence Convergence: ||s_n|-|L|| < epsilon

[tarheelborn]

Homework Statement

For any a, b in R, show that ||a|-|b|| <= |a-b|. Then prove that {|s_n|} converges to |L| if {s_n} converges to L.

Homework Equations

The Attempt at a Solution

For the first part, ||a|-|b|| = |a-b| by the triangle inequality. For the second part, ||s_n|-0| < epsilon implies that |s_n -0| < epsilon, but I am not sure how to work that around to the L's.

 

[g_edgar]

For the first part, ||a|-|b|| = |a-b| by the triangle inequality.

What? For example:
$\big|\,|2|-|-2|\,\big| = |2-(-2)|$
?

 

[fmam3]

Get the $$| |a| - |b| | \leq |a - b|$$ right first. The second part just follows by the Squeeze Theorem, or more simply, just "shove in the epsilons" if you know what I mean.